# Iterative Assessment and Improvement
of DNN Operational Accuracy

Antonio Guerriero, Roberto Pietrantuono, Stefano Russo  DIETI, Università
degli Studi di Napoli Federico II
Via Claudio 21, 80125 - Napoli, Italy
{antonio.guerriero, roberto.pietrantuono<EMAIL_ADDRESS>
###### Abstract

Deep Neural Networks (DNN) are nowadays largely adopted in many application
domains thanks to their human-like, or even superhuman, performance in
specific tasks. However, due to unpredictable/unconsidered operating
conditions, unexpected failures show up on field, making the performance of a
DNN in operation very different from the one estimated prior to release.

In the life cycle of DNN systems, the assessment of accuracy is typically
addressed in two ways: offline, via sampling of operational inputs, or online,
via pseudo-oracles. The former is considered more expensive due to the need
for manual labeling of the sampled inputs. The latter is automatic but less
accurate.

We believe that emerging iterative industrial-strength life cycle models for
Machine Learning systems, like MLOps, offer the possibility to leverage inputs
observed in operation not only to provide faithful estimates of a DNN
accuracy, but also to improve it through remodeling/retraining actions.

We propose DAIC (DNN Assessment and Improvement Cycle), an approach which
combines “low-cost” online pseudo-oracles and “high-cost” offline sampling
techniques to estimate and improve the operational accuracy of a DNN in the
iterations of its life cycle. Preliminary results show the benefits of
combining the two approaches and integrating them in the DNN life cycle.

###### Index Terms:

Deep Neural Networks, Accuracy assessment, Accuracy improvement

##  I Introduction

Nowadays, Machine Learning (ML) finds large adoption in various application
domains. This trend is due to the ability of ML, in particular of Deep Neural
Networks (DNN), to reach human beings’ effectiveness in many tasks [1, 2, 3].

The reliability of ML systems is usually measured in terms of accuracy. In the
case of classification, the accuracy is computed as the number of correctly
classified examples out of the total. The difficulty to automate the
assessment of DNN accuracy still represents a threat to their application also
in critical domains.

The main activities related to evaluating the accuracy and consequently
improving the DNN are typically executed before its release in the execution
environment. Metamorphic testing [4] and mutation testing [5, 6] represent the
most common strategies to evaluate the robustness of the DNN and to forecast
the reliability of these systems in the operational environment.

However, the accuracy estimated before release can substantially diverge from
the one obtained during operation (operational accuracy). Retch et al.
demonstrated how the accuracy scores of classifiers can significantly drop
when completely new data are submitted [7]. This problem grows up when
unexpected phenomena occur in operation, such as distribution shift or label
shift [8].

Iterative life cycles specific for DNN - such as MLOps [9, 10] \- have been
envisaged by companies like Google. In these DNN life cycle models,
development and operational stages are linked in a loop [11], aiming to assess
and improve the accuracy the DNN according to the operating conditions. In
particular, they may exploit operational data for remodeling and/or retraining
the DNN before the new deployment (experimental stage), and for both the
automatic evaluation of the accuracy and the automatic re-training of the
models on the field (deployment stage).

The true label of operational data collected by monitoring the DNN is
generally unknown. This is a general issue in software testing, known as the
oracle problem [12]. For ML systems, according to Murphy et al., “there is no
reliable test oracle to indicate what the correct output should be for
arbitrary input” [13]. The problem clearly affects also DNN accuracy
estimation [14].

Two approaches to assess DNN operational accuracy are:

  * i)

to automatically evaluate the correct classification of operational inputs by
means of pseudo-oracles (often in turn based on ML models), which may detect
mispredictions based on various sources of knowledge;

  * ii)

to reduce the size of the operational dataset to be labelled, by proper
statistical sampling of few representative inputs.

Pseudo-oracles do not need human intervention; however, they typically suffer
from a high number of false positives [15], due to the probabilistic nature of
the knowledge used to evaluate the output of the DNN under assessment.
Sampling techniques may reduce but not avoid costly and time-consuming
feedback from a human oracle; however, they avoid false positives, and provide
more faithful estimates of the DNN operational accuracy.

This paper proposes the DNN Assessment and Improvement Cycle (DAIC),
integrating automatic assessment via pseudo-oracles and assessment via
sampling. Its objectives are to provide faithful estimates of the operational
accuracy of a DNN while reducing the cost of manual intervention, and to
exploit the new labeled examples to take remodeling/retraining actions to
improve the DNN accuracy.

The preliminary results of experiments with the MNIST handwritten digits
dataset [16] show that DAIC is effective in providing DNN accuracy estimates,
by leveraging automatic pseudo-oracles to follow the accuracy of the DNN with
unlabeled samples, and triggering the high-cost sampling-based assessment only
when necessary to update estimates. Collected operational samples are then
further exploited to improve the accuracy of the DNN in an iteration cycle.
DAIC is robust to phenomena like label shift.

The paper is structured as follows. Section II describes the techniques for
the operational accuracy assessment of DNN. Section III introduces the DNN
Assessment and Improvement Cycle; Section IV presents the preliminary results.
Section V describes future plans; Section VI presents the conclusions.

##  II Operational accuracy assessment of DNN

###  II-A Assessment via pseudo-oracles

Automatic pseudo-oracles are typically built using cross referencing [17, 18,
19] based on the knowledge encoded into the training set. This knowledge is
extracted through multiple implementations – diverse from each other (e.g.,
different ML models, or same ML model but different architectures) - to
perform a majority voting. These techniques are strictly affected by biases in
the training set. When training data are not representative of the operational
environment, performance of that oracles degrades significantly.

Other techniques have been proposed to extract knowledge from the training
data to build automatic oracles, for instance, by using dedicated networks
(ConfidNet [20] and autoencoders [21]) or exploiting features of the system
under assessment itself (e.g. the output of internal layers [22]).

Techniques considering only the training dataset and the DNN as knowledge to
build a pseudo-oracle are particularly sensitive to deviations of the
operational context from the pre-deployment one. Therefore, they are expected
to poorly perform in presence of phenomena like label shift [8, 23].
Supervised DNN algorithms face a label shift when the distribution of the
labels of inputs changes with respect to training, despite everything else
remains unchanged: in practice, when unlabeled operational inputs are similar
to training examples, thus are classified by the DNN as per training, yet
their actual class is different from the one learnt during training.

For image classification problems, the ICOS oracle surrogate has been proposed
to assess the accuracy of Convolutional Neural Networks after their release in
operation [24]. ICOS extracts invariants form different sources of knowledge
to evaluate unlabeled operational examples.

Similarly to ICOS, we consider a pseudo-oracle – hereafter called DNN-OS -
which exploits three different sources of knowledge (the operational domain,
training data, and the DNN) to define three set of invariants (domain, data,
model), used to automatically evaluate the output of the DNN under assessment.

An example of domain invariant for an autonomous driving vehicle, assuming a
street with a speed limit of $50$ $km/h$, is: $fail\textrm{ :-
}speed\\_limit=50$ $km/h,accelerate=true,current\\_speed=50$ $km/h$. Such
domain invariants allow the oracle to detect failures looking at the output of
the DNN and at its effect on the whole system. The usage of domain invariants
makes DNN-OS robust against unexpected phenomena in operation with respect to
the state-of-the-art techniques.

Data and model invariants can be automatically extracted from the training and
validation datasets with a ML algorithm. These invariants look for patterns in
the input data and the DNN, respectively, such as a subset of pixels (for
data) or neurons (for the model) that always assume specific values when a
failure occurs.

For its characteristics, the assessment via pseudo-oracle can be performed
online, namely when the system is in operation. The automated oracle computes
the accuracy on actual inputs. This estimate can be used to suggest to the
testers if correcting/improving actions are needed.

The online assessment is characterized by a fixed cost for the “knowledge
extraction” and parameters tuning of the pseudo-oracle algorithm, which occur
una tantum.

###  II-B Assessment via sampling

The usage of sampling to reduce the cost of the manual labeling of operational
examples has been explored in the recent literature [25, 26, 27, 28]. Some
techniques are used to select a small data sample that accurately represents
the population [25, 26, 27] to obtain a faithful estimate of the accuracy
provided during operation. A representative sample would roughly contain the
same proportion of examples causing misprediction as the operational dataset.

However, the mere imitation of the expected input can be inefficient,
especially with very accurate DNN, because of the great effort to manually
label correctly classified examples to get an acceptable estimate of the
operational accuracy. This cost makes it evident that maximizing the sampling
of examples related to wrong outputs, while still getting an unbiased estimate
of the operational accuracy, is preferable.

With DeepEST, Guerriero et al. [28] aim to both provide faithful estimates of
the operational accuracy, but trying to sample more failures examples (e.g.
misclassifications) and balancing the unequal sampling during the estimation
process.

This assessment strategy can be performed offline, namely when the monitored
operational data are available together with the outputs of the DNN. With this
data, an estimate of the accuracy can be computed via sampling and manual
labeling. The high cost of manual labeling the operational input is balanced
by the possibility to use the labeled examples to take improving actions for
the DNN under assessment.

##  III DNN Assessment and Improvement Cycle

A way to reduce the cost of applying and maximize the benefit is to combine
the online and offline assessment in a cycle, called DNN Assessment and
Improvement Cycle (DAIC). The idea is to have at each cycle a “low-cost”
estimate of the accuracy provided through a pseudo-oracle, and to trigger a
“high-cost” (but more faithful) offline sampling-based estimate only when the
operational accuracy estimated by the automatic pseudo-oracle drops under a
given threshold.

Figure 1: The proposed MLOps-like DNN Assessment and Improvement Cycle (DAIC)

Like MLOps, DAIC entails an experimental and a deployment stage, with the
following phases (Figure 1):

  1. 1.

Data Preprocessing: in the starting phase of a cycle, the training and the
verification datasets are updated considering new labeled examples (if
available) and based on the accuracy estimate computed in the previous
iteration. This phase updates the training set so as to better represent the
operating conditions actually observed.

  2. 2.

Remodeling and Retraining: the model is trained from scratch (first iteration
or in case of re-modeling), or re-trained with the training set output of the
Data Processing phase.

  3. 3.

Model verification: the accuracy of the trained DNN is computed prior to
release (verification accuracy) on the verification dataset generated by the
Data Preprocessing phase.

  4. 4.

Deploy: the DNN is deployed into the execution environment, and put in
operation.

  5. 5.

Monitoring: (unlabeled) inputs to the DNN and the corresponding DNN outcomes
are collected; additional information on operating conditions (input sources,
user typologies, operational profile, etc.) may be collected, if available, to
build domain invariants.

  6. 6.

Assessment via pseudo-oracle: an automatic pseudo-oracle is used to classify
each output of the DNN as Pass or Fail. The oracle predictions are used to
compute an estimate of the DNN accuracy in operation, called predicted
accuracy.

  7. 7.

Evaluation: when the predicted and the verification accuracy diverge, an
offline assessment session is triggered (8); otherwise, the sampling-based
assessment is skipped.

  8. 8.

Assessment via sampling: a set of inputs is sampled and (manually) labeled,
and an estimate of the operational accuracy is computed.

The idea is to consider the pseudo-oracle for a continuous evaluation of the
operational accuracy provided by the DNN to reduce the cost of manual
labeling, retraining, and remodeling, performing them only when required.

##  IV Experiments

###  IV-A Accuracy assessment algorithms, datasets, open artifacts

DAIC experiments have been conducted with two pseudo-oracles and one sampling
accuracy assessment algorithms.

The two pseudo-oracles are SelfChecker [22] (the automatic oracle that
exploits the features of the DNN under test itself to evaluate the
predictions), and DNN-OS.

The sampling-based assessment algorithm is DeepEST [28]; it considers
auxiliary variables, such as the confidence of DNN predictions, to guide the
sampling through as much as possible failing examples and to balance the
unequal sampling in the estimation.

The dataset considered for the preliminary experiments is MNIST [16], a famous
dataset for handwritten digits classification. In particular, $1,000$ examples
are considered for training, $500$ for the verification set, and $1,000$
unlabeled inputs (for each cycle) as the operational dataset.

DNN-OS invariants are obtained as follows:

  * •

Domain invariants: defined by domain experts about the input sources of the
DNN. In particular, for MNIST, we assume that users insert input into three
different forms requiring respectively digits without straight lines {0, 3, 6,
8, 9}, digits with straight lines {1, 4, 7} only, and remaining digits {2, 5}.
We define an invariant for each form. The output provided by the DNN for each
operational input is checked against the set of possible digits expected for
the source form.

  * •

Data invariants: automatically extracted from the training data in form of
decision rules (C4.5 algorithm [29]) and filtered based on their confidence
($C\geq 0.99$) and support ($S\geq 10$).

  * •

Model invariants: extracted from validation data with Random Forest using the
output of the neurons of the last layer as features.

The sample size considered for DeepEST is $500$, and the proportion of
examples sampled randomly with respect to those with weighted sampling is set
to $0.5$.

For independent verification or replication, the experimental code is
available on GitHub at:

https://github.com/dessertlab/DAIC.git.

Figure 2: DAIC results with the SelfChecker pseudo-oracle TABLE I: DAIC detailed results with the SelfChecker pseudo-oracle | verification | actual | predicted acc. | estimated acc.
---|---|---|---|---
cycle | accuracy | accuracy | (SelfChecker) | (DeepEST)
1 | 0.861 | 0.859 | 0.881 | untriggered
2 | 0.861 | 0.863 | 0.883 | untriggered
3 | 0.861 | 0.860 | 0.879 | untriggered
4 | 0.861 | 0.700 | 0.868 | untriggered
5 | 0.861 | 0.698 | 0.861 | untriggered
6 | 0.861 | 0.707 | 0.866 | untriggered
7 | 0.861 | 0.695 | 0.864 | untriggered
8 | 0.861 | 0.692 | 0.867 | untriggered

###  IV-B Results

DAIC is experimented by running eight cycles, with five repetitions. The
pseudo-oracle assessment is executed at each iteration.

In the experiments, the triggering condition for the sampling-based assessment
is:

$\\{$predicted accuracy $<$ (verification accuracy \- $0.05$)$\\}$ OR

$\\{$predicted accuracy $<$ minimum accuracy$\\}$

that is, the offline assessment is triggered when the difference between the
accuracy estimated online (predicted accuracy) and the accuracy estimated
prior to release (verification accuracy) drops below a given threshold (here
set to $0.05$), or when the predicted accuracy falls below a minimum accuracy
required for the DNN (set to $0.80$ in the experiments).

When DeepEST is triggered, a set of new manually labeled samples are sent to
the Data Preprocessing phase, where they are integrated into the training and
verification sets. The proportion between new and old samples in the training
dataset may be varied according to the accuracy estimates of last cycle(s). By
default, both new and old samples are considered.

Figures 2 and 3 show the average results and the confidence intervals over 5
repetitions of 8 DAIC iterations. Tables I and II provide the details for each
cycle. The first three cycles represent the nominal conditions, namely when
the training and validation set faithfully represent the operational dataset.
As expected, the operational accuracy computed with SelfChecker (Figure 2, and
first three rows of Table I) and with DNN-OS (Figure 3, and first three rows
of Table II) does not trigger sampling in the first three cycles.

Figure 3: DAIC results with the DNN-OS pseudo-oracle TABLE II: DAIC detailed results with the DNN-OS pseudo-oracle | verification | actual | predicted acc. | estimated acc.
---|---|---|---|---
cycle | accuracy | accuracy | (DNN-OS) | (DeepEST)
1 | 0.861 | 0.859 | 0.833 | untriggered
2 | 0.861 | 0.863 | 0.828 | untriggered
3 | 0.861 | 0.860 | 0.825 | untriggered
4 | 0.861 | 0.700 | 0.678 | 0.713
5 | 0.833 | 0.726 | 0.693 | 0.719
6 | 0.810 | 0.781 | 0.725 | 0.784
7 | 0.818 | 0.888 | 0.850 | untriggered
8 | 0.818 | 0.894 | 0.851 | untriggered

Starting from the fourth cycle, a label shift is simulated by switching labels
2 and 7 in operational data. As SelfChecker relies only on model and data
information, it is unable to detect failures, and its accuracy estimate
diverges from the actual one without triggering the assessment via sampling
for the remaining cycles (Figure 2, and cycle 4 in Table I).

Thanks to domain invariants, DNN-OS is able to correctly detect failures,
providing an accurate estimate of the operational accuracy. Indeed, when the
accuracy drops (Figure 3), it triggers the assessment via sampling (cycle 4 in
Table II). DeepEST estimate confirms that the operational dataset is starting
to diverge from the one observed in the previous cycles. Thus, new samples are
inserted into the training dataset, and the model is re-trained. During cycles
5 and 6, the accuracy drops are still correctly detected by DNN-OS, and
further improvement actions are performed with the new samples provided by
DeepEST.

During the 7th cycle the model is trained only with the $1,500$ examples
collected in cycles 4, 5, and 6 achieving a high actual accuracy. In both
cycles 7 and 8 the accuracy is correctly estimated by DNN-OS as greater than
0.8, avoiding the triggering of DeepEST.

DNN-OS exhibits a single unnecessary trigger in cycle 6 (repetition 2), where
it does not catch that the actual accuracy was already higher than the
minimum.

##  V Future Plans

Life cycles for DNN-based systems adopted in continuous delivery contexts are
iterative by nature. The proposed integration of pseudo-oracles and sampling
techniques supports both the assessment and the improvement of the DNN
accuracy. It helps engineers to leverage collected features in the operational
environment to more faithfully evaluate and then specialize the DNN performing
the task they need in the way they need.

We plan to refine DAIC defining more sophisticated strategies for the
automatic improvement of the DNN in the loop. Techniques like DeepEST can spot
a high number of failing examples, which, along with operational features, can
help improve the performance of DNN also in corner cases. An advancement is to
integrate the automatic improvement both at the experimental and deployment
stage. As shown in the preliminary results, it is rarely required to change a
well-performing model in case of unexpected phenomena like label shift. Often,
additional training or training from scratch, by incorporating the operational
examples in the training set, may suffice to improve the operational accuracy.
For this reason, in line with MLOps perspectives, strategies for the online
auto-improvement of DNN can be based on the “probabilistic” output of the
pseudo-oracles. Moreover, by automating data preprocessing, the offline re-
training step can be run without human intervention.

To a second extent, we plan to apply inferential engines on operational
features to automatically extract operational constraints aiming to improve
pseudo-oracle effectiveness in estimating the accuracy during the operation. A
recent work from Google stresses the importance of incorporating domain
knowledge as a set of rules to improve ML components accuracy [30].

The iterative assessment and the improvement of the accuracy of the DNN can be
of interest beyond the experimented image classification domain. We plan to
apply DAIC in industry-relevant domains like Autonomous Driving: for instance,
to the throttle/braking/steering angle prediction, which are regression
problems.

##  VI Conclusions

Preliminary results confirm that the accuracy computed before the release can
be very different from the one achieved in operation by the DNN in presence of
unexpected phenomena like label shift. However, the accuracy predicted by DNN-
OS follows the actual accuracy with the operational data thanks the domain
invariants, triggering the assessment via sampling only when required.

The estimates provided by DeepEST can be used to faithfully evaluate the
accuracy provided in operation. The experimental results also showed that the
performance of the DNN in operation can be sensibly increased thanks to the
availability of the new labeled examples.

## Acknowledgment

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 871342 “uDEVOPS”. It is also supported by the DIETI COSMIC
project.

## References

  * [1] N. Kühl, M. Goutier, L. Baier, C. Wolff, and D. Martin, “Human vs. supervised machine learning: Who learns patterns faster?” _CoRR_ , vol. abs/2012.03661, 2020. [Online]. Available: https://arxiv.org/abs/2012.03661
  * [2] K. He, X. Zhang, S. Ren, and J. Sun, “Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification,” in _International Conference on Computer Vision (ICCV)_. IEEE, 2015, pp. 1026–1034.
  * [3] D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. Driessche, T. Graepel, and D. Hassabis, “Mastering the game of go without human knowledge,” _Nature_ , vol. 550, pp. 354–359, 10 2017.
  * [4] X. Xie, L. Ma, F. Juefei-Xu, M. Xue, H. Chen, Y. Liu, J. Zhao, B. Li, J. Yin, and S. See, “DeepHunter: A Coverage-Guided Fuzz Testing Framework for Deep Neural Networks,” in _28th ACM SIGSOFT Int. Symposium on Software Testing and Analysis (ISSTA)_. ACM, 2019, pp. 146–157.
  * [5] L. Ma, F. Zhang, J. Sun, M. Xue, B. Li, F. Juefei-Xu, C. Xie, L. Li, Y. Liu, J. Zhao, and Y. Wang, “DeepMutation: Mutation Testing of Deep Learning Systems,” in _29th International Symposium on Software Reliability Engineering (ISSRE)_. IEEE, 2018, pp. 100–111.
  * [6] Y. Li, W. Shen, T. Wu, L. Chen, D. Wu, Y. Zhou, and B. Xu, “How higher order mutant testing performs for deep learning models: A fine-grained evaluation of test effectiveness and efficiency improved from second-order mutant-classification tuples,” _Information and Software Technology_ , vol. 150, 2022.
  * [7] B. Recht, R. Roelofs, L. Schmidt, and V. Shankar, “Do ImageNet Classifiers Generalize to ImageNet?” in _Proceedings of Machine Learning Research (PMLR)_ , K. Chaudhuri and R. Salakhutdinov, Eds., vol. 97, 2019, pp. 5389–5400.
  * [8] S. Garg, Y. Wu, S. Balakrishnan, and Z. C. Lipton, “A unified view of label shift estimation,” in _34th Conference on Neural Information Processing System (NeurIPS)_ , 2020.
  * [9] S. Alla and S. K. Adari, “What Is MLOps?” in _Beginning MLOps with MLFlow: Deploy Models in AWS SageMaker, Google Cloud, and Microsoft Azure_. Berkeley, CA: Apress, 2021, pp. 79–124.
  * [10] Google, “MLOps: Continuous delivery and automation pipelines in machine learning,” Available: http://cloud.google.com/solutions/machine-learning/mlops-continuous-delivery-and-automation-pipelines-in-machine-learning. [Online].
  * [11] R. Ashmore, R. Calinescu, and C. Paterson, “Assuring the Machine Learning Lifecycle: Desiderata, Methods, and Challenges,” _ACM Computing Surveys_ , vol. 54, no. 5, 2021.
  * [12] E. T. Barr, M. Harman, P. McMinn, M. Shahbaz, and S. Yoo, “The Oracle Problem in Software Testing: A Survey,” _IEEE Transactions on Software Engineering_ , vol. 41, no. 5, pp. 507–525, 2015.
  * [13] C. Murphy, G. Kaiser, L. Hu, and L. Wu, “Properties of Machine Learning Applications for Use in Metamorphic Testing,” in _20th International Conference on Software Engineering and Knowledge Engineering (SEKE)_ , 2008, pp. 867–872.
  * [14] A. Guerriero, “Reliability Evaluation of ML systems, the oracle problem,” in _IEEE International Symposium on Software Reliability Engineering Workshops (ISSREW)_ , 2020, pp. 127–130.
  * [15] J. Zhang, M. Harman, L. Ma, and Y. Liu, “Machine learning testing: Survey, landscapes and horizons,” _IEEE Transactions on Software Engineering_ , vol. 48, no. 1, pp. 1–36, 2022.
  * [16] Y. LeCun and C. Cortes, “MNIST handwritten digit database,” http://yann.lecun.com/exdb/mnist/, 2010.
  * [17] S. Srisakaokul, Z. Wu, A. Astorga, O. Alebiosu, and T. Xie, “Multiple-implementation testing of supervised learning software,” in _AAAI Workshops_. Association for the Advancement of Artificial Intelligence, 2018.
  * [18] K. Pei, Y. Cao, J. Yang, and S. Jana, “DeepXplore: Automated Whitebox Testing of Deep Learning Systems,” _Communications of the ACM_ , vol. 62, no. 11, p. 137–145, 2019.
  * [19] H. Wang, J. Xu, C. Xu, X. Ma, and J. Lu, “Dissector: Input Validation for Deep Learning Applications by Crossing-Layer Dissection,” in _42nd International Conference on Software Engineering (ICSE)_. ACM, 2020, p. 727–738.
  * [20] C. Corbière, N. Thome, A. Bar-Hen, M. Cord, and P. Pérez, “Addressing failure prediction by learning model confidence,” in _Advances in Neural Information Processing Systems_ , H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, Eds., vol. 32. Curran Associates, Inc., 2019.
  * [21] A. Stocco, M. Weiss, M. Calzana, and P. Tonella, “Misbehaviour prediction for autonomous driving systems,” in _42nd International Conference on Software Engineering (ICSE)_. ACM, 2020, p. 359–371.
  * [22] Y. Xiao, I. Beschastnikh, D. S. Rosenblum, C. Sun, S. G. Elbaum, Y. Lin, and J. S. Dong, “Self-checking deep neural networks in deployment,” in _43rd International Conference on Software Engineering (ICSE)_. IEEE, 2021, pp. 372–384.
  * [23] J. Quiñonero-Candela, M. Sugiyama, A. Schwaighofer, and N. D. Lawrence, _When Training and Test Sets Are Different: Characterizing Learning Transfer_. MIT Press, 2009, pp. 3–28.
  * [24] A. Guerriero, M. R. Lyu, R. Pietrantuono, and S. Russo, “Assessing Operational Accuracy of CNN-based Image Classifiers using an Oracle Surrogate,” _Intelligent Systems with Applications_ , vol. 17, pp. 1–13, 2023.
  * [25] Z. Li, X. Ma, C. Xu, C. Cao, J. Xu, and J. Lü, “Boosting Operational DNN Testing Efficiency through Conditioning,” in _Proc. 27th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering (ESEC/FSE)_. ACM, 2019, pp. 499–509.
  * [26] J. Chen, Z. Wu, Z. Wang, H. You, L. Zhang, and M. Yan, “Practical accuracy estimation for efficient deep neural network testing,” _ACM Transactions on Software Engineering and Methodology_ , vol. 29, no. 4, 2020.
  * [27] X. Gao, Y. Feng, Y. Yin, Z. Liu, Z. Chen, and B. Xu, “Adaptive test selection for deep neural networks,” in _44th International Conference on Software Engineering (ICSE)_. ACM, 2022, p. 73–85.
  * [28] A. Guerriero, R. Pietrantuono, and S. Russo, “Operation is the Hardest Teacher: Estimating DNN Accuracy Looking for Mispredictions,” in _43rd International Conference on Software Engineering (ICSE)_. IEEE, 2021, pp. 348–358.
  * [29] J. R. Quinlan, _C4.5: Programs for Machine Learning_. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc., 1993\.
  * [30] A. Choudhary, “Google AI researchers present a new method to train models, ‘DeepCTRL’,” Analytics India Magazine (online), February 13, 2022, https://analyticsindiamag.com/google-ai-researchers-present-a-new-method-to-train-models-deepctrl/, Accessed: 2022-10-03.


# BCMCC
Exercises 3

Marcel Fernandez and Jorge J. Urroz<EMAIL_ADDRESS><EMAIL_ADDRESS>
# A Study of the Separating Property in Reed-Solomon Codes by Bounding the
Minimum Distance

Marcel Fernandez and Jorge J. Urroz<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

According to their strength, the tracing properties of a code can be
categorized as frameproof, separating, IPP and TA. It is known that if the
minimum distance of the code is larger than a certain threshold then the TA
property implies the rest. Silverberg et al. ask if there is some kind of
tracing capability left when the minimum distance falls below the threshold.
Under different assumptions, several papers have given a negative answer to
the question. In this paper further progress is made. We establish values of
the minimum distance for which Reed-Solomon codes do not posses the separating
property.

###### Key words and phrases:

Reed-Solomon codes IPP codes Separating codes

##  1\. Introduction.

As a motivation for our work, consider the distribution of digital goods. In
the trade of digital content, safe guarding ownership rights is certainly a
critical issue. A way to protect copyright consists of making each copy of the
content unique. This is done by embedding a different mark in each delivered
item. These hidden marks are typically strings of symbols. However, since now
all objects are different, traitor users can get together and by comparing
their copies, they create a new copy that tries to disguise their identities.
This is known as a collusion attack and the newly created copy is usually
called a pirate copy.

A way to deal with collusion attacks is by taking the embedded symbol strings
to be the code words of a code with tracing properties. There is a large
literature about codes possessing different degrees of robustness against
collusion attacks. Let us give a brief overview. Formal definitions will be
done in subsequent sections. In a $c$-frameproof code [3], a coalition of at
most $c$ users can not create a pirate copy that contains the code word of
another user not in the coalition. In $c$-secure frameproof codes two disjoint
coalitions of at most $c$ users can not create the same pirate copy. It has
been shown [12], that the secure frameproof property is the same as the
separating property [14]. Codes with the Identifiable Parent Property (IPP)
were introduced in [10]. Informally, a code has the $c$-IPP property if all
coalitions of at most $c$ traitors that can generate the same pirate copy have
a non-empty intersection, i.e. have a common user. The IPP has received
considerable attention in the recent years, having been studied by several
authors [1, 2, 17, 4, 9]. An even stronger property is the Traceability
property ($c$-TA). In this case, it is guaranteed that the “closest”
authorized copy to a given pirate copy belongs to one of the traitors.
Sufficient conditions for a code to be a $c$-TA code are stated in [16].

The work in [15] discusses efficient algorithms for the identification of
traitors in schemes that use $c$-TA codes. Let $M$ denote the size of the
code. For TA codes, tracing is an $O(M)$ process, whereas for IPP codes
tracing is more expensive since it is an $O(\binom{M}{c})$ process. Being the
TA property stronger than the IPP, but being tracing more costly for the IPP,
it seems reasonable to expect that by relaxing the TA requirements one is left
with a code that, even though is no longer $c$-TA, still possesses IPP. In
this regard, Silverberg et al. asked the following question:

Question 1 [15]: Is it the case that all $c$-IPP Reed-Solomon codes are also
$c$-TA?

Although intuition might lead us to give a negative answer, in that same paper
the authors used truncated Reed-Solomon codes to credit the exact opposite,
that is, if a Reed-Solomon code does not have the TA property then it neither
has the IPP. Later, the work in [13] not only reinforced this conjecture, but
proved a stronger fact, a Reed-Solomon code that is not $c$-TA it is neither
$c$-secure-frameproof. Therefore, they generalized the above question to the
following one:

Question 2 [13]: Is it the case that all $c$-SEP Reed-Solomon codes are also
$c$-TA?

In this paper, we supplement more evidence to this last question. The results
we present will hopefully contribute to a complete understanding of the
tracing properties in Reed-Solomon codes.

##  2\. Definitions and previous results.

Let $q$ be a prime power $p^{s}$ and let $\mathbb{F}_{q}$ denote the finite
field with $q$ elements. Denoting by $\mathbb{F}_{q}^{n}$ the set of all
$n$-tuples with elements from $\mathbb{F}_{q}$. We define a linear code of
length $n$ to be a vector subspace of $\mathbb{F}_{q}^{n}$. Then,
$\mathbb{F}_{q}$ s is called the code alphabet, and the $n$-vectors in the
code are called code words. The dimension of the code is defined as the
dimension of the vector subspace. Let
$\mathbf{u},\mathbf{v}\in{\rm{I\\!F}}_{q}^{n}$ be two words, then the Hamming
distance ${\it d}(\mathbf{u},\mathbf{v})$ between $\mathbf{u}$ and
$\mathbf{v}$ is the number of positions where $\mathbf{u}$ and $\mathbf{v}$
differ. The minimum distance $d$, is defined as the smallest distance between
two different code words. A linear code with length $n$, dimension $k$ and
minimum distance $d$ is denoted as a $[n,k,d]$-code.

Reed-Solomon codes can be defined as follows. Let $\mathbb{F}_{q}[x]$ be the
ring of polynomials over $\mathbb{F}_{q}$. Take all polynomials of degree less
than $k$, $\mathbb{F}_{q}[x]_{k-1}\subset\mathbb{F}_{q}[x]$. Let $\alpha$ be a
primitive element of $\mathbb{F}_{q}$, so we have
$\\{1,\alpha,\alpha^{2},\ldots,\alpha^{q-2}\\}=\mathbb{F}_{q}\setminus\\{0\\}$.

######  Definition 1.

A Reed-Solomon, $RS[n,k]_{q}$, code is defined as the vector subspace of
$\mathbb{F}_{q}^{n}$ determined by all vectors of the form

$\mathbf{v}=(f(0),f(1),f(\alpha),\ldots,f(\alpha^{q-2}))$

where $f\in\mathbb{F}_{q}[x]_{k-1}$. Note that $n=q$.

As in the previous definition, throughout the paper, and probably with an
slight abuse of notation we will denote polynomials with an italic lowercase
letter.

Reed-Solomon codes are maximum distance separable (MDS) [11]. That means they
attain the Singleton bound with equality $d=n-k+1$.

###  2.1. Definitions about codes with tracing properties.

Let $C$ be an $[n,k,d]$ code over $\mathbb{F}_{q}$, let
$T=\\{\mathbf{t}^{1},\ldots,\mathbf{t}^{c}\\}\subseteq C$ with
$\mathbf{t}^{i}=(t_{1}^{i},\ldots,t_{n}^{i})$ be a subset of size $c$. Also,
let $T|_{i}=\\{t_{i}^{j}|\mathbf{t}^{j}\in T\\}$. The _descendant set_ of $T$,
is defined as

$desc(T)=\left\\{\mathbf{z}=(z_{1},\ldots,z_{n})\in\mathbb{F}_{q}^{n}|z_{i}\in
T|_{i}\\},1\leq i\leq n\right\\}.$

######  Definition 2.

A code $C$, defined over $\mathbb{F}_{q}$, has the $c$-separating property
(denoted $c$-SEP), $c>0$, if for any two disjoint subsets of $C$,
$U=(\mathbf{u}^{1},\ldots,\mathbf{u}^{c})$ and
$V=(\mathbf{v}^{1},\ldots,\mathbf{v}^{c})$, we have

$U|_{i}\cap V|_{i}=\emptyset\ \ \mathrm{for}\ \ 1\leq i\leq n.$

In the introduction we used the name secure frame-proof for the separating
property.

######  Definition 3.

A code $C$, defined over $\mathbb{F}_{q}$, has the $c$-Identifiable Parent
Property (denoted $c$-IPP), $c>0$, if for all
$\mathbf{z}\in\mathbb{F}_{q}^{n}$ and for all coalitions $T\subseteq C$ of at
most $c$ code words, we have

$\mathbf{z}\not\in\bigcup_{T,|T|\leq c}desc(T)\ \ \mathrm{or}\ \
\bigcap_{\mathbf{z}\in desc(T)}T\neq\emptyset.$

######  Definition 4.

A code $C$ is a $c$-traceability code (denoted $c$-TA), for $c>0$, if for all
subsets (coalitions) $T\subseteq C$ of at most $c$ code words, if
$\mathbf{z}\in desc(T)$, then there exists a $\mathbf{t}\in T$ such that
$d(\mathbf{z},\mathbf{t})<d(\mathbf{z},\mathbf{w})$ for all $\mathbf{w}\in
C-T$.

We will also have ocasion to link our discussion to a weaker tracing property
called $c$-frameproof (FP).

######  Definition 5.

A code $C$, defined over $\mathbb{F}_{q}$, has the $c$-Frameproof Property
(denoted $c$-FP), $c>0$, if for any code word $\mathbf{u}$ and a subset of $C$
of size at most $c$ $V=(\mathbf{v}^{1},\ldots,\mathbf{v}^{c})$, with
$\mathbf{u}\notin V$, we have

$\mathbf{u}|_{i}\notin V|_{i}\ \ \mathrm{for}\ \ 1\leq i\leq n.$

Note that $(c,1)$-SEP is equivalent to $c$-FP.

###  2.2. Bezout identity

Some of the results in this paper, make extensive use of the Bezout identity.
Intuitively, the Bezout identity is the ability to do the euclidean algorithm
backwards.

######  Definition 6.

Let $u$ and $v$ two elements in a ring $A$, and let $d=(a,b)$ a greatest
common divisor of $a$ and $b$. The Bezout identity is an identity of the form

$au-by=d,$

for some elements $a$ and $b$ in the ring $A$.

###  2.3. The separating condition for Reed-Solomon codes.

Let us state previous results that lead to the motivation of our work. In [16,
Lemma 1.6] authors show that if $|C|>c\geq q$ then $C$ is not a $c$-IPP code.
In [16, Lemma 1.3] it is shown that the $c$-TA property implies the $c$-IPP
property. In [5, 6][16, Theorem 4.4] it is proved that any $[n,k,d]$ code over
$\mathbb{F}_{q}$ with $d>n-n/c^{2}$ and $q>c$ is a $c$-TA code.

Interestingly enough in [15, Theorem 8] authors construct a family of
truncated ($n<q-1$) $RS[n,k]_{q}$ codes that fail to be $c$-IPP if
$c^{2}>n/(n-d)$. Then in [15, Question 11] the authors ask if it is always
true that for Reed-Solomon codes the $c$-IPP fails if $c^{2}>n/(n-d)$.

This is a very interesting question because a positive answer would mean that
for Reed Solomon the $c$-IPP and $c$-TA properties are essentially the same.

This question has been addressed in [8, 13]. We can summarize previous results

######  Theorem 1 ([8], Theorem 6).

Let $RS[n,k]_{q}$ be a Reed-Solomon code over $\mathbb{F}_{q}$ such that $k-1$
divides $q-1$. Then, if $d\leq n-\frac{n}{c^{2}}$ the code is not $c$-SEP.

######  Corollary 1 ([13], Corollary 2).

Let $C$ be a $[n,k,d]$ Reed-Solomon code over $\mathbb{F}_{q}$. If
$c\geq\sqrt{q-1}$ and $d\leq(1-1/c^{2})n$, then the code is not $c$-SEP.

######  Theorem 2 ([13],Theorem 2).

Let $RS[n,k,d]_{q}$ be a Reed-Solomon code over $\mathbb{F}_{q}$ and $c$ a
divisor of $q$. Then, if $d\leq n-\frac{n}{c^{2}}$ the code is not $c$-SEP.

It is worth noting that the proofs of these theorems are constructive in the
sense that explicit disjoint sets $U,V$, such that

$U\cap V=\emptyset\ \ \mathrm{and}\ \ desc(U)\cap desc(V)=\emptyset$

are found. Therefore proving that for Reed-Solomon codes under the conditions
of the theorems $c$-SEP, $c$-IPP and $c$-TA are in fact equivalent.

###  2.4. Our contribution

In this paper, progress in the understanding of the tracing properties in
Reed-Solomon codes is made. In the flavour of Theorem 1 and Theorem 2 we use
the structure of the finite field $\mathbb{F}_{q}$, over which the code is
defined. In our particular case, we take advantage of the divisors of $q-1$.
With that, we are able to give a complete answer to Question 2 by proving, in
a constructive way, that in Reed-Solomon codes $c$-IPP and $c$-TA properties
are essentially the same when $q\equiv 1\pmod{c^{2}}$. More precisely, we set
the minimum distance to $d=\lceil n-\frac{n}{c^{2}}\rceil$ which is the
maximum allowed so the code is not $c$-TA and then find two disjoint sets of
code words that are not separated. In the rest of the paper, although the
proofs are also constructive, the approach is somehow different. We relax the
distance condition and study whether a $[n,k,d]$ Reed-Solomon code over
$\mathbb{F}_{q}$ with minimum distance $d<(n-r)$ is $c$-SEP for some
$r>n/c^{2}$. To show the flavour of our approach, we start by proving a
relaxed version of the question and state that Reed-Solomon codes are not
$c$-SEP for $r=n/c$ . Then, we proceed to strengthen this result. For the case
$c=2$, we answer the question for $r=\left[\frac{q}{3}\right]$ and for $c=3$
we do so for $r=2\left[\frac{q}{8}\right]$. We round up the paper using an
elegant result of Cilleruelo [7] to give an alternate and more concise proof
of known results.

##  3\. A connection with the frameproof property.

We start our discussion by studying Reed-Solomon codes over $\mathbb{F}_{q}$
with minimum distance $d\leq q-\frac{q}{c}$.

######  Theorem 3.

For any $q$ a power of prime, $c\geq 2$ and $d\leq q-\dfrac{q}{c}$, Reed-
Solomon $[n,k,d]$ codes over $\mathbb{F}_{q}$ are not $c$-SEP.

###### Proof.

Let $\mathbb{F}_{q}=\\{\alpha_{1},\alpha_{2}\dots,\alpha_{q}\\}$ and $c\geq 2$
an integer. To construct the Reed Solomon code we consider any set of $c$
distinct polynomials $\\{f_{1},\dots,f_{c}\\}\subset\mathbb{F}_{q}[x]_{k-1}$
for $k-1=q-d$. Observe that in this case $c(k-1)\geq q$.

Let $q=cl+r$ where $0\leq r<c$. Since the case $c|q$ is taken care of in
Theorem 2, we consider $c\nmid q$ and then $r>0$. We take the maximum allowed
minimum distance $d=\lfloor q-\frac{q}{c}\rfloor$. Then,
$l=\frac{q}{c}-\frac{r}{c}<k-1$ and since $l$ is an integer $l\leq k-2$ which
means $l+1\leq k-1$.

For $1\leq i\leq r$ we take

$g_{i}=f_{i}+\prod_{m=1}^{l+1}(x-\alpha_{(l+1)(i-1)+m}),$

while for $r+1\leq i\leq c$ take

$g_{i}=f_{i}+\prod_{m=1}^{l}(x-\alpha_{(i-1)l+m+r}).$

Observe that $g_{i}\in\mathbb{F}_{q}[x]_{k-1}$ for all $1\leq i\leq c$. Then,
by construction, for every $1\leq j\leq q$ the polynomials $f_{t}$ and $g_{t}$
with $t=[\frac{j}{k-1}]$ are such that $f_{t}(\alpha_{j})=g_{t}(\alpha_{j})$.
Note that $t\leq j/(k-1)\leq q/(k-1)\leq c$. Hence
$\\{f_{1}(\alpha),\dots,f_{c}(\alpha)\\}\cap\\{g_{1}(\alpha),\dots,g_{c}(\alpha)\\}\neq\emptyset$
for any $\alpha\in\mathbb{F}_{q}$ and so the two sets citeof code words given
by $(f_{i}(\alpha_{1}),\dots,f_{i}(\alpha_{n})$ and
$(g_{i}(\alpha_{1}),\dots,g_{i}(\alpha_{n})$ for $1\leq i\leq c$ are not
separated.

We finally show that the constructed polynomials are all different. Observe
that for any two polynomials $f,g$ of degree smaller than $q$ it is not
possible that $f(\alpha)=g(\alpha)$ for every $\alpha\in\mathbb{F}_{q}$, since
in that case $(x^{q}-x)|(f-g)$. Hence, taking $f_{i}\neq f_{j}$ for any $i\neq
j$, then for any $1\leq j,l\leq c$ there exist an $\alpha\in\mathbb{F}_{q}$
such that $g_{j}(\alpha)\neq f_{l}(\alpha)$ and also for any for $i\neq j$,
there exist a $\beta\in\mathbb{F}_{q}$ such that $g_{i}(\beta)\neq
g_{j}(\beta)$ since for any two polynomials in the code $f,g$ we have
deg$(f-g)\leq k-1=q-d<q$.

Finally, note that, since the code is MDS, taking $n=q$ we have $d=q-k+1$ and
the condition $d\leq q-q/c^{2}$ implies $q\leq c^{2}(k-1)$ which is true in
our case since $c^{2}(k-1)\geq c(k-1)\geq q$. ∎∎

In Section 2.1 we defined $c$-frameproof codes. Take an $[n,k,d]_{q}$ code. It
is well known, see proof of Lemma III.2 in [3], that if $d>n-\frac{n}{c}$ then
the code is $c$-FP. In Theorem 3 we have also proved that a Reed-Solomon code
with minimum distance $d\leq n-\frac{n}{c}$ is not $c$-FP. Indeed, by taking
$f_{1}=f_{2}=\cdots=f_{c}=\alpha$, $\alpha\in\mathbb{F}_{q}$ in the proof of
the theorem, for every $\alpha\in\mathbb{F}_{q}$ there is a $g_{i}$ such that
$g_{i}(\alpha)=\alpha$.

##  4\. Increasing the minimum distance.

In the previous section we saw that a Reed Solomon code with a small distance
is not separated. This is consistent with intuition since then the code has a
higher dimension as a vector space then chances are code words are not
“separated”. In this section we discuss strategies to increase the minimum
distance of the code and still keep non-separation.

###  4.1. The case $c=2$.

To show our approach we first deal with a particular case.

######  Lemma 1.

The $[11,4,8]$ Reed-Solomon code over $F_{11}$ is not $(2,2)$-separating.

###### Proof.

We will find polynomials $f_{1}$,$f_{2}$ and $g_{1}$,$g_{2}$ such that the
corresponding pairs of codewords $\\{\mathbf{f}_{1},\mathbf{f}_{2}\\}$ and
$\\{\mathbf{g}_{1},\mathbf{g}_{2}\\}$ are not separated.

Consider the polynomial $f_{1}=0$, and take
$g_{1}=\gamma_{1}\prod_{i=1}^{3}(x-\alpha_{i})$, for some
$\\{\gamma_{1},\alpha_{1},\alpha_{2},\alpha_{3}\\}\in\mathbb{F}_{11}$. Now,
let

$f_{2}=\sum_{i=1}^{3}g_{1}(\alpha_{3+i})\frac{\prod_{j\in\\{1,2,3\\},j\neq
i}(x-\alpha_{3+j})}{\prod_{j\in\\{1,2,3\\},j\neq
i}(\alpha_{3+i}-\alpha_{3+j})}+\phi_{2}\prod_{i=1}^{3}(x-\alpha_{3+i}),$

for some $\\{\phi_{2},\alpha_{4},\alpha_{5},\alpha_{6}\\}\in\mathbb{F}_{11}$.
Finally, consider

$g_{2}=\gamma_{2}\prod_{i=1}^{3}(x-\alpha_{6+i}),$

for some
$\\{\gamma_{2},\alpha_{7},\alpha_{8},\alpha_{9}\\}\in\mathbb{F}_{11}$.

By construction
$\\{f_{1}(\alpha_{i}),f_{2}(\alpha_{i})\\}\cap\\{g_{1}(\alpha_{i}),g_{2}(\alpha_{i})\\}\neq\emptyset$
for $i=1,\dots,9$. Now, selecting $\phi_{2},\gamma_{2}$ such that

$\displaystyle\sum_{i=1}^{3}g_{1}(\alpha_{3+i})\frac{\prod_{j\in\\{1,2,3\\},j\neq
i}(\alpha_{10}-\alpha_{3+j})}{\prod_{j\in\\{1,2,3\\},j\neq
i}(\alpha_{3+i}-\alpha_{3+j})}$ $\displaystyle=$
$\displaystyle\gamma_{2}\prod_{i=1}^{3}(\alpha_{10}-\alpha_{6+i})-\phi_{2}\prod_{i=1}^{3}(\alpha_{10}-\alpha_{3+i})$
$\displaystyle\sum_{i=1}^{3}g_{1}(\alpha_{3+i})\frac{\prod_{j\in\\{1,2,3\\},j\neq
i}(\alpha_{11}-\alpha_{3+j})}{\prod_{j\in\\{1,2,3\\},j\neq
i}(\alpha_{3+i}-\alpha_{3+j})}$ $\displaystyle=$
$\displaystyle\gamma_{2}\prod_{i=1}^{3}(\alpha_{11}-\alpha_{6+i})-\phi_{2}\prod_{i=1}^{3}(\alpha_{11}-\alpha_{3+i}),$

which is possible whenever

$\prod_{i=1}^{3}(\alpha_{10}-\alpha_{6+i})(\alpha_{11}-\alpha_{3+i})\neq\prod_{i=1}^{3}(\alpha_{11}-\alpha_{6+i})(\alpha_{10}-\alpha_{3+i}),$

we get

$\displaystyle f_{2}(\alpha_{10})$ $\displaystyle=$ $\displaystyle
g_{2}(\alpha_{10})$ $\displaystyle f_{2}(\alpha_{11})$ $\displaystyle=$
$\displaystyle g_{2}(\alpha_{11})$

and hence the pairs $\\{\mathbf{f}_{1},\mathbf{f}_{2}\\}$ and
$\\{\mathbf{g}_{1},\mathbf{g}_{2}\\}$ are not separated. ∎∎

As an example take $\alpha_{i}=i$. Then

$\prod_{i=1}^{3}(\alpha_{10}-\alpha_{6+i})(\alpha_{11}-\alpha_{3+i})=\prod_{i=1}^{3}(4-i)(8-i)=7!/4,$

while

$\prod_{i=1}^{3}(\alpha_{11}-\alpha_{6+i})(\alpha_{10}-\alpha_{3+i})=\prod_{i=1}^{3}(5-i)(7-i)=4\cdot
6!.$

In this case

$\displaystyle g_{1}$ $\displaystyle=$ $\displaystyle x^{3}-6x^{2}+11x-6$
$\displaystyle f_{2}$ $\displaystyle=$ $\displaystyle 5x^{3}+10x+9$
$\displaystyle g_{2}$ $\displaystyle=$ $\displaystyle 10x^{3}+x^{2}+x+9$

and we have that

$\displaystyle\mathbf{f}_{1}$ $\displaystyle=$
$\displaystyle(0,0,0,0,0,0,0,0,0,0,0)$ $\displaystyle\mathbf{f}_{2}$
$\displaystyle=$ $\displaystyle(2,3,9,6,2,5,1,9,4,5,9)$
$\displaystyle\mathbf{g}_{1}$ $\displaystyle=$
$\displaystyle(0,0,0,6,2,5,10,1,6,9,5)$ $\displaystyle\mathbf{g}_{2}$
$\displaystyle=$ $\displaystyle(6,1,10,5,2,6,0,0,0,5,9)$

are not separated. It is clear that the solution is not unique, Taking
$\alpha_{i}=i+1$, we get

$\displaystyle g_{1}$ $\displaystyle=$ $\displaystyle x^{3}-9x^{2}+26x-24$
$\displaystyle f_{2}$ $\displaystyle=$
$\displaystyle-\frac{13}{4}x^{3}+\frac{135}{2}x^{2}-\frac{1715}{4}x+\frac{1737}{2}$
$\displaystyle g_{2}$ $\displaystyle=$ $\displaystyle-x^{3}+27x^{2}-242x+720,$

which give the pairs

$\displaystyle\mathbf{f}_{1}$ $\displaystyle=$
$\displaystyle(0,0,0,0,0,0,0,0,0,0,0)$ $\displaystyle\mathbf{f}_{2}$
$\displaystyle=$ $\displaystyle(9,2,3,9,6,2,5,1,9,4,5)$
$\displaystyle\mathbf{g}_{1}$ $\displaystyle=$
$\displaystyle(5,0,0,0,6,2,5,10,1,6,9)$ $\displaystyle\mathbf{g}_{2}$
$\displaystyle=$ $\displaystyle(9,6,1,10,5,2,6,0,0,0,5)$

that are also not separated.

Note that the previous lemma answers Question 2 for $q=11$ and $c=2$ and for
these particular values improves Theorem 3. However this approach does not
fully generalize. Fortunately, for general $q$ we can give a fully explicit
answer, by decreasing a bit the distance of the code.

######  Theorem 4.

A Reed Solomon code over $\mathbb{F}_{q}$ with distance
$d=q-\left[\frac{q}{3}\right]$ is not $2$-SEP.

###### Proof.

Let $q=3l+r$ where $0\leq r\leq 2$, and consider for any $1\leq i,j\leq 2$,
$A_{i,j}\subset\mathbb{F}_{q}$ disjoint sets such that $A_{2,2}=\emptyset$,
and $|A_{i,j}|=l+1$ for $r$ of them and $|A_{i,j}|=l$ for the remaining $3-r$
sets. Now let $p_{i,j}=\prod_{\alpha\in A_{i,j}}(x-\alpha)$. Observe that
$\cup_{i,j}A_{i,j}=\mathbb{F}_{q}$. The following polynomials, of degree at
most $l+1$ define a code that is not $2$-SEP.

$\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle
f_{2}$ $\displaystyle=$ $\displaystyle p_{2,1}-p_{1,1},$ $\displaystyle g_{1}$
$\displaystyle=$ $\displaystyle-p_{1,1},$ $\displaystyle g_{2}$
$\displaystyle=$ $\displaystyle-p_{1,2}.$

Indeed, for any $\alpha\in A_{1,1}$ we have $f_{1}=g_{1}$, for those
$\alpha\in A_{1,2}$, $f_{2}=g_{2}$, and for those $\alpha\in A_{2,1}$,
$f_{2}=g_{1}$, completing all the roots in $\mathbb{F}_{q}$. ∎∎

###  4.2. The case $c=3$.

Le us move to a larger value of $c$ and deal with the case $c=3$.

This representation of the Bezout identity in Definition 6 is not unique. For
univariate polynomials we have the following lemma. We include proofs for
clarity in the exposition.

######  Lemma 2.

Let $u,v\in\mathbb{F}_{q}[x]$ be two non constant polynomials relatively
prime. Then there exist polynomials $a,b\in\mathbb{F}_{q}[x]$ such that

$1=au-bv,$

with $\deg(a)<\deg(v)$ and $\deg(b)<\deg(u)$.

###### Proof.

By the Bezout identity we have that, for some $\hat{a},\hat{b}\in F_{q}[x]$,

(1) $\hat{a}u-\hat{b}v=1$

If $\deg(\hat{a})<\deg(v)$ then the theorem follows because

(2) $\deg(\hat{a})+\deg(u)=\deg(\hat{b})+\deg(v).$

Assume $\deg(\hat{a})\geq\deg(v)$. It is clear that the pair $a=\hat{a}+tv$,
$b=\hat{b}+tu$ also satisfy Bezout’s identity for any $t\in\mathbb{F}_{q}[x]$.
Dividing $\hat{a}$ by $v$,we get $\hat{a}=q_{a}v+r_{a}$ with
$\deg(r_{a})<\deg(v)$ and taking $t=-q_{a}$, we have that $a=r_{a}$,
$b=\hat{b}-q_{a}u$ and the result follows using the same reasoning as in (2),
since $a$ and $b$ also satisfy Bezout’s identity. ∎∎

######  Corollary 2.

Let $u,v\in\mathbb{F}_{q}[x]$ be two non constant polynomials relatively
prime. Let $z$ a polynomial such that $\deg(z)<\deg(u)+\deg(v)$. Then, we can
express $z$ as

$z=au-bv,$

for $a,b\in\mathbb{F}_{q}[x]$with $\deg(a)<\deg(v)$ and $\deg(b)<\deg(u)$.

###### Proof.

By the previous lemma we have that there exist $\hat{a}$ and $\hat{b}$, such
that

(3) $\hat{a}u-\hat{b}v=1$

with $\deg(\hat{a})<\deg(v)$ and $\deg(\hat{b})<\deg(u)$. Therefore,

(4) $\hat{a}zu-\hat{b}zv=z$

It is clear that

$\displaystyle a$ $\displaystyle=\hat{a}z+tv$ (5) $\displaystyle b$
$\displaystyle=\hat{b}z+tu,$

satisfy

(6) $au-bv=z,$

for any $t\in\mathbb{F}_{q}[x]$.

Now, since we can express $\hat{a}z=vq_{a}+r_{a}$, we substitute $t=-q_{a}$ in
(4.2) obtaining,

$\displaystyle a$ $\displaystyle=r_{a}$ (7) $\displaystyle b$
$\displaystyle=\hat{b}z-q_{a}u.$

The polynomials $a$ and $b$ in (4.2) satisfy (6). We have the following cases.
If $\deg(r_{a}u)\leq\deg(z)$ then $\deg(bv)\leq\deg(z)$ and so
$\deg(b)\leq\deg(z)-\deg(v)<\deg(u)$. On the other hand, if
$\deg(r_{a}u)>\deg(z)$ then we have $\deg(r_{a}u)=\deg(bv)$ and so
$0<\deg(v)-\deg(a)=\deg(u)-\deg(b)$. ∎∎

We express our result in the form of a theorem.

######  Theorem 5.

Let $q$ be a power of a prime, $c=3$ and $d<q-2\left[\frac{q}{8}\right]$. A
Reed Solomon code of length $q$ over $\mathbb{F}_{q}$ with distance $d$ is not
$3$-separating.

###### Proof.

Let $q=8l+r$ where $0\leq r<8$. We make a partition of $\mathbb{F}_{q}$ in
nine disjoint sets $U_{i,j}\subset\mathbb{F}_{q}$, $1\leq i,j\leq 3$, as
follows: $U_{1,1}=\emptyset$, $r$ sets of size $l+1$ and the other remaining
$8-r$ sets of size $l$. Observe that $\sum_{i,j}|U_{i,j}|=q.$

Now let $u_{1,1}=-1$, $u_{i,j}=\prod_{\alpha\in U_{i,j}}(x-\alpha)$ for
$(i,j)\neq(1,1)$ and take $v_{i,j}$ for $i\neq 1\neq j$ the solutions of
smaller degree of the Bezout equations

$\displaystyle v_{3,3}u_{3,3}-v_{3,2}u_{3,2}=1,$ $\displaystyle
v_{2,3}u_{2,3}-v_{2,2}u_{2,2}=1.$

Note that according to Corollary 2, the degree of $v_{2,2}$, $v_{2,3}$,
$v_{3,2}$ and $v_{3,3}$ is less than $l+1$.

Then we proceed similarly to define $v_{2,1},v_{3,1},v_{1,2}$ and $v_{1,3}$ to
be the solutions of smaller degree of the Bezout equations

$\displaystyle v_{3,3}u_{3,3}-v_{2,3}u_{2,3}=v_{3,1}u_{3,1}-v_{2,1}u_{2,1}$
$\displaystyle v_{3,3}u_{3,3}-v_{3,2}u_{3,2}=v_{1,3}u_{1,3}-v_{1,2}u_{1,2}.$

Again, by Corollary 2 the degree of $v_{2,1},v_{3,1},v_{1,2}$ and $v_{1,3}$ is
less than $l+1$. Finally, take
$v_{1,1}=v_{3,3}u_{3,3}-v_{3,1}u_{3,1}-v_{1,3}u_{1,3}.$ Now we define

(8) $\begin{array}[]{lc}g_{1}=-v_{3,1}u_{3,1},&\quad
f_{1}=g_{1}+v_{1,1}u_{1,1},\\\ g_{2}=-v_{3,2}u_{3,2},&\quad
f_{2}=g_{2}+v_{2,2}u_{2,2},\\\ g_{3}=-v_{3,3}u_{3,3},&\quad
f_{3}=g_{3}+v_{3,3}u_{3,3}.\end{array}$

By definition, we have

$(x^{q}-x)\left|\prod_{1\leq i,j\leq 3}v_{i,j}u_{i,j}=\prod_{1\leq i,j\leq
3}(f_{i}-g_{j}).\right.$

By construction we see that deg$(f_{i})\leq 2l+1$, deg$(g_{i})\leq 2l+1$ and
hence the result follows. ∎∎

##  5\. The general case.

In order to obtain stronger results we need to deal with larger values of both
$c$ and the minimum distance. The following theorem generalizes Theorem 4 for
$c\geq 2$.

######  Theorem 6.

Let $q$ be a power of a prime, $c\geq 2$ and
$d<q-\left[\frac{q}{2c-1}\right]$. A Reed Solomon code over $\mathbb{F}_{q}$
with distance $d$ is not $c$-SEP.

###### Proof.

Let $q=(2c-1)l+r$ where $0\leq r<2c-1$, and consider for any $1\leq i\leq
2c-1$, $A_{i}\subset\mathbb{F}_{q}$ disjoint sets such that: $r$ of the sets
are of size $|A_{i}|=l+1$ and the remaining $2c-r-1$ are of size $|A_{i}|=l$.
Now let $p_{i}=\prod_{\alpha\in A_{i}}(x-\alpha)$. Observe that
$\cup_{i}A_{i}=\mathbb{F}_{q}$. The following polynomials, of degree at most
$l+1$ evaluate to code words of a code that is not $c$-SEP.

$\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle
f_{i+1}$ $\displaystyle=$ $\displaystyle p_{c+i}-p_{i},\quad\text{ for }1\leq
i\leq c-1$ $\displaystyle g_{i}$ $\displaystyle=$ $\displaystyle-
p_{i},\quad\text{ for }1\leq i\leq c.$

Indeed, observe that $f_{1}(\alpha)=g_{i}(\alpha)$ for any $\alpha\in
A_{i},1\leq i\leq c$, while $f_{i}(\alpha)=g_{i}(\alpha)$ for any $\alpha\in
A_{c+i}$ for any $2\leq i\leq c$. ∎∎

To cope with a larger minimum distance, we would like to extend Theorem 5.
Unfortunately, the generalization is not immediate because when $c$ grows, the
degree of the polynomials $v_{i,j}$ blows up. To obtain stronger results we
need to take advantage of the structure of the field over which the code is
defined. In this case we are able to state a result for a minimum distance
matching the conjectured one, but limited to certain parameters of the code.

######  Theorem 7.

Let $m^{2}|q-1$. Then, for any $c\geq m$, there exist a non extended Reed
Solomon code over $F_{q}$ with distance $d=q-\frac{q-1}{m^{2}}$ that is not
$c$-SEP.

Proof. Let $\alpha$ be a primitive root of the multiplicative group
$\mathbb{F}_{q}^{*}$. Consider $g_{i}=\alpha^{i\frac{(q-1)}{d^{2}}}$, and
$f_{i}=\alpha^{\frac{-i(q-1)}{d}}x^{\frac{q-1}{d^{2}}}$, $i=0,\dots,c-1$. Now,
every element of $\mathbb{F}_{q}^{*}$ can be written as
$\alpha_{r,s}=\alpha^{ld^{2}+rd+s}$ for some $0\leq s,r<d$, and certain
integer $l$. Then we have clearly $f_{r}(\alpha_{r,s})=g_{s}$, proving the
result.

######  Corollary 3.

Let $c$ be any integer and $q\equiv 1\pmod{c^{2}}$. There exist a non extended
Reed Solomon code with distance $d=q-\left[\frac{q}{c^{2}}\right]$ over
$F_{q}$ which is $(c,c)-$ inseparable.

Proof. Simply note that if $c^{2}|q-1$, then
$\frac{q-1}{c^{2}}=\left[\frac{q}{c^{2}}\right]$

######  Corollary 4.

For any $p$ and $c$ there exist infinitely many $q=p^{e}$ such that
$\mathbb{F}_{q}$ admits a non extended Reed-Solomon code of distance
$d=q-\left[\frac{q}{c^{2}}\right]$.

Proof. Simply note that by Fermat’s little theorem $p^{\varphi(c^{2})}\equiv
1\pmod{c^{2}}$, so the result follows for any $e=k\varphi(c^{2})$,
$k\in\mathbb{N}$, applying the previous theorem.

##  6\. The “linear” case $d=q-1$

The case presented in this section is already dealt with in Corollary 1. We
include it here, because the proofs provide new ways to approach a complete
solution to the problem.

The first result we prove is a straight forward application of the following
theorem of J. Cilleruelo.

######  Theorem 8.

(Cilleruelo) Let $\alpha$ be a generator of $\mathbb{F}_{q}^{*}$. Then

$\\{\alpha^{i}-\alpha^{j}\,:\,0\leq i,j\leq 2q^{3/4}\\}=\mathbb{F}_{q}.$

Now, we have

######  Theorem 9.

Let $c\geq 2q^{3/4}$. Then, $[n,k,d]$ Reed Solomon codes over $F_{q}$ of
length $n$ and distance $d=q-1$ are not $c$-SEP.

Proof. Let us note first that, since $c\geq 2q^{3/4}$, then $c^{2}>q$ and then
$q-[q/c^{2}]-1=q-1$. So we need a code with distance $d=q-1$. This means that
we need to find two families of polynomials of size $c$ each, with all
polynomials of degree at most $1$, such that

$\\{f_{1}(\alpha),\dots,f_{c}(\alpha)\\}\cap\\{g_{1}(\alpha),\dots,g_{c}(\alpha)\\}\neq\emptyset$

for any $\alpha\in\mathbb{F}_{q}$. This is the same as

$(x^{q}-x)\left|\prod_{1\leq i,j\leq c}(f_{i}-g_{j}).\right.$

Now, let $\alpha$ be a generator of $\mathbb{F}_{q}^{*}$, and consider
$f_{i}=x-\alpha^{i}$, $g_{i}=-\alpha^{i}$, for $i=1,\dots,c$. It follows that,

$\prod_{1\leq i,j\leq c}(f_{i}-g_{j})=\prod_{1\leq i,j\leq
c}(x-(\alpha^{i}-\alpha^{j}))$

and by the previous theorem we trivially have

$(x^{q}-x)\left|\prod_{1\leq i,j\leq c}(f_{i}-g_{j}).\right.$

But we can make it better.

######  Theorem 10.

Suppose that $q-1=rs$ such that $(r,s)=1$ and suppose $c>\max\\{r,s\\}$. Then,
$[n,k,d]$ Reed Solomon codes with distance $d=q-1$ over $F_{q}$ which is not
$c$-SEP.

Proof. Let $q-1=rs$ such that $(r,s)=1$, $\alpha$ a generator of
$\mathbb{F}_{q}$ and consider the sets
$A=\\{1,\alpha^{r},\dots,\alpha^{r(s-1)}\\}$ and
$B=\\{1,\alpha^{s},\dots,\alpha^{s(r-1)}\\}$. Then, all the quotients $a/b$
with $a\in A$ and $b\in B$ are distinct. Indeed, suppose
$\alpha^{ri}/\alpha^{sj}=\alpha^{rI}/\alpha^{sJ}$. Then
$\alpha^{r(i-I)}/\alpha^{s(J-j)}$, and so $\alpha^{r(i-I)-s(J-j)}=1$ but,
since $\alpha$ is a generator, this is only possible either if
$r(i-I)-s(J-j)=0$ or else if $(q-1)|r(i-I)-s(J-j)$. In any of the two cases,
since $r|q-1$, we have $r|s(J-j)$ and since $(r,s)=1$, then $r|(J-j)$ but this
is impossible, since $|J-j|<r$, unless $J=j$, and then $i=I$.

Now, consider polynomials $f_{i}=\alpha^{ri}x$, $g_{j}=\alpha^{sj}$ with
$0\leq i\leq s-1$ and $0\leq j\leq r-1$. We can do that since $l<c$. By the
previous argument, the roots of $f_{i}-g_{j}$ are all distinct and we have
$rs=q-1$ distinct roots. Since $r<c$ we can just add the root missing by
adding a polynomial $g_{r}$.

Observe that, since $c>s$, then $c^{2}>rs=q-1$, and we can suppose $c^{2}>q$
since the case $c^{2}=q$ is already proved. So $[q/c^{2}]=0$ and the correct
distance is $d=q-1$, so we have to consider linear polynomials. (By the
theorem of catalan $q^{x}-p^{y}=1$ only if $3^{2}-2^{3}$)

The previous theorem improves Theorem 9 when $q$ is an even power. Indeed, in
the case in which $q=p^{2t}$, then either $(p^{t}-1)/2$ is odd or
$(p^{t}+1)/2$ is odd. Without loss of generality, assume $(p^{t}-1)/2$ is odd.
Then, we can take $r=(p^{t}-1)/2$ and $s=2(p^{t}+1)$ and so $s\leq 4r+4$.
Therefore $q-1=rs\leq 4r^{2}+4r$ or $q\leq 4r^{2}+4r+1=(2r+1)^{2}$ which gives
$r\geq(\sqrt{q}-1)/2$. Then, $q-1=rs\geq((\sqrt{q}-1)/2)s$ which gives
$s\leq\frac{q-1}{(\sqrt{q}-1)/2}=2(\sqrt{q}+1)$. Hence, since Theorem 10
assumes $c>s$ then for any $c>2(\sqrt{q}+1)$ we have that Reed Solomon code
with distance $d=q-1$ over $F_{q}$ are not $c$-SEP, improving Theorem 9.

In general, the theorem provides a general bound on $c$, depending on the
factorization on the exponent. However, in the case of a sophie germain prime,
$q-1=2p$ where q and $p$ are primes, then Theorem 10 only gives $c\geq q/2$.

##  7\. Conclusion

The aim of the paper is to find out whether or not there exist values of the
minimum distance for which a Reed-Solomon is $c$-SEP but not $c$-TA. We start
the presentation by considering a sufficiently small value to the minimum
distance. For this much convenient value, we prove that codes do not posses
the frameproof property, let alone the separating one. For the cases $c=2$ and
$c=3$, we improve this almost naive result by introducing to our discourse
both polynomial interpolation and Bezout’s identity.

The approach for case $c=3$ does not generalize to larger values of $c$. In
order to deal with the general case, we resort to the structure of
$\mathbb{F}_{q}$, the finite field over which the code is defined. This allows
us to prove an assertion for all $c$, whenever $q\equiv 1\pmod{c^{2}}$. Along
the same line of reasoning, we provide an alternative proof of existing
results by applying an elegant theorem about the generator of the
multiplicative group of $\mathbb{F}_{q}$.

Our presentation shows that for the general case, a constructive proof is by
no means trivial. This is because, when using the structure of the field
defining the code one can not encircle all cases and cases without “structure”
do not seem to follow any common pattern. So, although the problem is
algebraic in nature, it seems that an existence proof should be considered.

## References

  * [1] A. Barg, G. Cohen, S. Encheva, G. Kabatiansky, and G. Zémor.  A hypergraph approach to the identifying parent property: the case of multiple parents.  Technical report, DIMACS 2000-20, 2000.
  * [2] Alexander Barg and Gregory A. Kabatiansky.  A class of i.p.p. codes with efficient identification.  J. Complexity, 20(2-3):137–147, 2004.
  * [3] D. Boneh and J. Shaw.  Collusion-secure fingerprinting for digital data.  IEEE Trans. Inform. Theory, 44(5):1897–1905, 1998.
  * [4] Minquan Cheng, Hung-Lin Fu, Jing Jiang, Yuan-Hsun Lo, and Ying Miao.  Codes with the identifiable parent property for multimedia fingerprinting.  Designs, Codes and Cryptography, 83, 11 2014.
  * [5] B. Chor, A. Fiat, and M. Naor.  Tracing traitors.  Advances in Cryptology-Crypto’94, LNCS, 839:480–491, 1994.
  * [6] B. Chor, A. Fiat, M. Naor, and B. Pinkas.  Tracing traitors.  IEEE Trans. Inform. Theory, 46:893–910, 2000.
  * [7] J. Cilleruelo.  Combinatorial problems in finite fields and Sidon sets.  Combinatorica, 32(5):497–511, 2012.
  * [8] Marcel Fernandez, Josep Cotrina, Miguel Soriano, and Neus Domingo.  A note about the identifier parent property in reed-solomon codes.  Comput. Secur., 29(5):628–635, 2010.
  * [9] Yujie Gu, Minquan Cheng, Grigory Kabatiansky, and Ying Miao.  Probabilistic Existence Results for Parent-Identifying Schemes.  arXiv e-prints, page arXiv:1906.01031, Jun 2019.
  * [10] Henk D. L. Hollmann, Jack H. van Lint, Jean-Paul Linnartz, and Ludo M. G. M. Tolhuizen.  On codes with the Identifiable Parent Property.  J. Combinatorial Theory, 82(2):121–133, May 1998.
  * [11] F. J. MacWilliams and N. J. A. Sloane.  The Theory of Error-Correcting Codes.  North Holland, 1977.
  * [12] José Moreira, Marcel Fernández, and Grigory Kabatiansky.  Almost separating and almost secure frameproof codes over q-ary alphabets.  Des. Codes Cryptogr., 80(1):11–28, 2016.
  * [13] José Moreira, Marcel Fernandez, and Miguel Soriano.  On the relationship between the traceability properties of reed-solomon codes.  Adv. Math. Commun., 6(4):467–478, 2012.
  * [14] Yurii L’vovich Sagalovich.  Separating systems.  Problems Inform. Transmission, 30(2):105–123, 1994.
  * [15] A. Silverberg, J. Staddon, and J.L. Walker.  Applications of list decoding to tracing traitors.  IEEE Trans. Inform. Theory, 49(5):1312–1318, May 2003.
  * [16] Jessica N Staddon, Douglas R Stinson, and Ruizhong Wei.  Combinatorial properties of frameproof and traceability codes.  IEEE Transactions on Information Theory, 47(3):1042–1049, 2001\.
  * [17] Tran van Trung and Sosina Martirosyan.  New constructions for ipp codes.  Des. Codes Cryptography, 35(2):227–239, 2005.


# Chiral Dirac Equation and Its Spacetime and CPT Symmetries

Timothy B. Watson  University of Texas at Arlington  Zdzislaw E. Musielak
University of Texas at Arlington

###### Abstract

The Dirac equation with chiral symmetry is derived using the irreducible
representations of the Poincaré group, the Lagrangian formalism, and a novel
method of projection operators that takes as its starting point the minimal
assumption of four linearly independent physical states. We thereby
demonstrate the fundamental nature of this form of the Dirac equation. The
resulting equation is then examined within the context of spacetime and CPT
symmetries with a discussion of the implications for the general formulation
of physical theories.

##  1 Introduction

Since its introduction in 1928 numerous modifications of the Dirac operator
[1] have been proposed resulting in a number of so-called generalized Dirac
equations. Such generalizations have been invoked for the purposes of unifying
leptons and quarks [2-4], accounting for the three families of elementary
particles [5-9], including ad hoc a pseudoscalar mass [10], and extending the
Dirac equation to distances comparable to the Planck length [11]. One most
recent modification of the Dirac equation has included the effects of chiral
symmetry [12]. In this generalization, the most general first order partial
differential equation for bispinors constrained via Poincaré invariance is
derived, resulting in the chiral Dirac equation (CDE) which differs from the
original by an additional degree of freedom in the form of the chiral angle.
Specific solutions to this equation necessitate a specific choice of chiral
basis. These results have been used to explain the smallness of neutrino
masses and some properties of dark matter [12].

With the benefit of hindsight and the accumulation of experimental evidence,
it may be said that one of the great insights of Driac was the observation
that elementary particles exhibit a quantization of both angular momentum
(spin-up/spin-down) and sign of the energy (matter/antimatter). Modern
particle physics has reinforced this picture for free particles of a given
flavor independent of internal (gauge) symmetries. In this work, we
demonstrate how the assumption of two quantized characteristics is equivalent
to the chiral form of the Dirac equation (CDE), Dirac’s original equation
being a special case.

In an attempt to demonstrate these result holistically, the present paper is
organized as follows: Section 2 presents three different methods of deriving
the CDE operator beginning with the standard group theory and Lagrange
formalism derivations before presenting a novel derivation from the assumption
of the existence of independent physical states and their corresponding
orthogonal idempotents (projection operators) paramaterized by physically
meaningful quantities; Section 3 then sets out a detailed investigation into
the spacetime (continuous) and CPT (discrete) symmetries underlying the
derived CDE, their role and constraints in the derivation procedures, and the
physical implications resulting from this equation; Section 4 is devoted to
conclusions.

##  2 Methods to Derive Chiral Dirac Equation

###  2.1 Group Theoretical Derivation

To derive the Dirac equation with chiral symmetry, we begin by following
Wigner [13,14] and identifying particles with the induced irreducible
representations (irreps) of the Poincaré group $\mathcal{P}\ =\
SO(3,1)\otimes_{s}T(3+1)$, with $SO(3,1)$ being a non-invariant Lorentz group
of rotations and boosts and $T(3+1)$ an invariant subgroup of spacetime
translations. The condition that the Dirac spinor wavefunction transforms as
one of the irreps of $T(3+1)\subset\mathcal{P}$ can be written as the
following matrix eigenvalue equation [12]

$X^{\mu}\partial_{\mu}\psi=-Y\psi\ ,$ (1)

where $\mu$ = $0$, $1$, $2$ and $3$ and

$\psi=\begin{bmatrix}\chi_{L}\\\ \chi_{R}\\\ \end{bmatrix}$ (2)

is the Dirac spinor wavefunction with $\chi_{L}$ and $\chi_{R}$ being two-
component bispinors. Here $X^{\mu}$ and $Y$ are five $4\times 4$ which
nontrivially mix the spinor components. To determine the form of these
matrices, we constrain them to satisfy covariant conditions under the Lorentz
transformations of SO(3,1), and find the resulting Poincaré invariant chiral
Dirac equation (CDE) [12] to be given by

$(i\gamma^{\mu}\partial_{\mu}-me^{i\alpha\gamma^{5}})\psi=0\ ,$ (3)

where $\alpha$ is the chiral angle. This angle appears as a necessary degree
of freedom as it is unconstrained by the covariant transformation constraints.
It may be noted also that this degree of freedom is a consequence of taking
our bispinors to be eigenstates of CPT symmetry and not eigenstates of parity
as is conventionally required in deriving the Dirac equation 111We thank M.
Efroimsky for bringing this point to our attention [15].

Because this method of deriving the CDE is based explicitly on the symmetries
of Minkowski spacetime, we may be certain the resulting equation is Poincaré
invariant. Furthermore, The CDE correctly accounts for all four sets of states
(spins up and down, matter and antimatter), and its solutions have important
physical implications [12]. Being Poincaré invariant and local, the CDE
satisfies the two basic characteristics required for the equation to be
fundamental. Two other characteristics are gauge invariance and the existence
of Lagrangian. Since we consider only free elementary particles, gauge
invariance is not discussed here. We now demonstrate that the CDE can be
formally derived from Lagrangian formalism, which is sufficient to call the
CDE the fundamental equation of physics.

###  2.2 Derivation from Lagrangian formalism

The Lagrangian formalism is a powerful and independent way to derive a
dynamical equation. The Lagrangian for the Dirac equation ($\alpha=0$ in Eq.
3) is very well-known and presented in textbooks (e.g., [15,16]) without
derivation. In fact, the Lagrangian was not a part of the Dirac’s original
paper where the equation first appeared [1]. An interesting attempt to obtain
the Dirac Lagrangian is presented and discussed in [17]. Let us briefly review
the main points of this attempt and then use them to obtain the Lagrangian for
Eq. (3).

In case $\alpha=0$, Eq. (3) reduces to the Dirac equation

$(i\gamma^{\mu}\partial_{\mu}-m)\psi=0\ ,$ (4)

which describes a free, massive, non-chiral and spin 1/2 relativistic
elementary particle [1, 15-17].

To obtain the Lagrangian density for this equation, we follow [17] and require
that the Lagrangian is a hermitian, single-valued proper scalar or pseudo-
scalar in $\psi$ and $\partial_{\mu}\psi$. Since $\psi$ has the double-valued
properties under rotations, the terms in the Lagrangians must have even
numbers of $\psi$. The simplest proper scalar is $\bar{\psi}\psi$, where
$\bar{\psi}$ is the Dirac adjoint. Now, the construction of a scalar kinetic
term has to be done with caution as $\partial_{\mu}\psi$ requires saturation
of the index $\mu$, which cannot be done by another derivative since the
result would be a second-order equation. Therefore, the Dirac matrices
$\gamma^{\mu}$ are used to saturate the index $\mu$, and write the kinetic
term as $i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi$. Since the physical units
of the kinetic term are different than $\bar{\psi}\psi$, the latter must be
multiplied by an inverse length dimension, which in natural units is mass.
Then, the Dirac Lagrangian can be written in the following form

${\mathcal{L}_{D}}={1\over
2}\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi-{1\over
2}\bar{\psi}(i\gamma^{\mu}{\overleftarrow{\partial_{\mu}}}+m)\psi=0\ .$ (5)

This is a fully symmetric form of the Lagrangian, which shows that when
evaluated along a stationary path the Dirac Lagrangian vanishes [17]. Both the
Dirac equation and its Lagrangian are Poincaré invariant.

Using the above procedure, the Lagrangian for the CDE (see Eq. 3) can also be
obtained and written as

${\mathcal{L}_{CD}}={1\over
2}\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-me^{-i\alpha\gamma^{5}})\psi-{1\over
2}\bar{\psi}(i\gamma^{\mu}{\overleftarrow{\partial_{\mu}}}+me^{-i\alpha\gamma^{5}})\psi=0\
.$ (6)

Similarly to ${\mathcal{L}_{D}}$, the Lagrangian is also fully symmetric,
hermitian and single-valued proper scalar or pseudo-scalar, and it vanishes
when evaluated along a stationary path. Moreover, the CDE and its Lagrangian
are Galilean invariant. By substituting ${\mathcal{L}_{CD}}$ into the Euler-
Lagrange equation for variations with respect to $\bar{\psi}$, the CDE given
by Eq. (3) is obtained. This method of deriving the CDE is independent from
the group theory derivation and serves to demonstrate that the equation
satisfies a least-action principle requisite of any fundamental theory. Our
derivation based on projection operators is now presented.

###  2.3 Derivation from Orthogonal Idempotents

We may define a set of projection operators operating on an $N$-dimensional
complex vector space with any set of $N$-by-$N$ orthogonal idempotent matrices
satisfying

$\hat{P}_{i}\hat{P}_{j}=\hat{P}_{j}\hat{P}_{i}=\delta_{ij}\hat{P}_{i}\qquad\sum^{N}_{i=1}\hat{P}_{i}=1\
.$ (7)

The total number of projection operators of a given vector space is maximally
equal to the dimensions of the space considered. Therefore, for $N=2$, we may
expand the most general operators acting on a spinor in terms of the Pauli
matrices and the two-by-two identity matrix. Let

$\hat{P}_{1}=a_{0}I_{2}+\vec{\sigma}\cdot\vec{a}\qquad\hat{P}_{2}=b_{0}I_{2}+\vec{\sigma}\cdot\vec{b}\
.$ (8)

Then enforcing the orthogonal idempotent conditions, we obtain the following
constraints

$a_{0}=b_{0}=\frac{1}{2}\qquad\vec{a}\cdot\vec{a}=\vec{b}\cdot\vec{b}=\frac{1}{4}\qquad\vec{a}=-\vec{b}\
.$ (9)

Solving this we find two projection operators for our symmetry group whose
degrees of freedom may be parameterized in terms of the unit vector $\hat{a}$.
We write these projection operators succinctly as

$\hat{P}_{\pm}(\hat{a})=\frac{1}{2}(I_{2}\pm\vec{\sigma}\cdot\hat{a})\ .$ (10)

For a fixed $\hat{a}$, these operators allow us to define two types of objects
in our two-dimensional vector space. For any such element
$\chi\in\mathbb{C}^{2}$ we may define
$\chi_{\pm}(\vec{a})\equiv\hat{P}_{\pm}(\hat{a})\chi(\vec{a})$. It necessarily
follows that

$(\vec{\sigma}\cdot\hat{a})\chi_{\pm}(\vec{a})=\pm\chi_{\pm}(\vec{a})\ .$ (11)

It is now a simple matter to extend these projection operators to projections
in $2^{N}$-dimensional vector spaces. In general we may write

$\hat{P}_{s_{1},s_{2},...s_{N}}(\hat{a}_{1},\hat{a}_{2},...\hat{a}_{N})=\bigotimes_{i=1}^{N}\hat{P}_{s_{i}}(\hat{a_{i}})\
,$ (12)

for $s_{i}\in\\{\pm\\}$. It is easy to see that these projection operators
satisfy our orthogonal idempotent constraints. We then define our set of
eigenvectors in a similar manner

$\chi_{s_{1},s_{2},...s_{N}}(\vec{a}_{1},\vec{a}_{2},...\vec{a}_{N})=\bigotimes_{i=1}^{N}\chi_{s_{i}}(\vec{a_{i}})\
.$ (13)

By restricting our considerations to the vector space of spinors we are able
to give these abstract considerations physical significance. Recall that the
rotation operator corresponding to a rotation of $\theta$ about the axis
defined by $\hat{a}$ for the vector space of two-component spinors takes the
form

$\displaystyle\hat{R}(\theta,\hat{a})$
$\displaystyle=\cos{\frac{\theta}{2}}+i(\vec{\sigma}\cdot\hat{a})\sin{\frac{\theta}{2}}$
(14)
$\displaystyle=e^{\frac{i\theta}{2}}\hat{P}_{+}(\hat{a})+e^{-\frac{i\theta}{2}}\hat{P}_{-}(\hat{a})\
.$

It follows that eigenstates of our projection operators
$\hat{P}_{\pm}(\hat{a})$ are physically invariant under rotations about
$\hat{a}$, differing only by a phase. We therefore identify
$\hat{P}_{\pm}(\hat{a})$ as projecting out the portion of the state vector
with spin parallel ($+$) or anti-parallel ($-$) to the $\hat{a}$-axis. By
choosing $\vec{a}=\vec{p}$, where $\vec{p}$ is the three-momentum of the
particle, we find $\hat{P}_{\pm}(\hat{p})$ to be the helicity projection
operators. This most neatly encapsulates the experimental observance of binary
spin states in mathematical terms.

We now wish to use our projection operator methodology to classify states of
positive and negative energies, e.g. matter/anti-matter. The inclusion of an
additional two-valued quantum property necessitates (at minimum) a four-
dimensional vector space. We therefore construct the projection operators of
the form

$\hat{P}_{s_{1},s_{2}}(\hat{q},\hat{p})=\hat{P}_{s_{1}}(\hat{q})\otimes\hat{P}_{s_{2}}(\hat{p})\
,$ (15)

where we have introduced the vector $\hat{q}$ about which we will have more to
say shortly, for the time being $\hat{q}$ is simply a set of three complex
numbers and satisfies $\hat{q}\cdot\hat{q}=1$.

Next we define the operand

$\chi_{s_{1},s_{2}}(\hat{q},\hat{p})=\chi_{s_{1}}(\hat{q})\otimes\chi_{s_{2}}(\hat{p})\
.$ (16)

The corresponding generalization of Eq. (11) yields

$\displaystyle(\vec{\sigma}\cdot\hat{q}\otimes
I_{2})\chi_{s_{1},s_{2}}(\hat{q},\hat{p})=s_{2}\chi_{s_{2},r_{2}}(\hat{q},\hat{p})$
(17)
$\displaystyle(I_{2}\otimes\vec{\sigma}\cdot\hat{p})\chi_{s_{1},s_{2}}(\hat{q},\hat{p})=s_{1}\chi_{s_{1},s_{2}}(\hat{q},\hat{p})\
.$ (18)

Exploiting the fact that $s_{1}=\pm s_{2}$ we may construct the equations

$\displaystyle(\vec{\sigma}\cdot\hat{q}\otimes
I_{2}+I_{2}\otimes\vec{\sigma}\cdot\hat{p})\chi_{\pm,\mp}(\hat{q},\hat{p})=(\vec{\sigma}\cdot\hat{q}\oplus\vec{\sigma}\cdot\hat{p})\chi_{\pm,\mp}(\hat{q},\hat{p})=0$
(19) $\displaystyle(\vec{\sigma}\cdot\hat{q}\otimes
I_{2}-I_{2}\otimes\vec{\sigma}\cdot\hat{p})\chi_{\pm,\pm}(\hat{q},\hat{p})=(\vec{\sigma}\cdot\hat{q}\ominus\vec{\sigma}\cdot\hat{p})\chi_{\pm,\pm}(\hat{q},\hat{p})=0\
.$ (20)

We now identify the set of gamma matrices in the chiral representation

$\displaystyle\gamma^{0}$ $\displaystyle=\sigma^{1}\otimes I_{2}$
$\displaystyle\gamma^{k}$ $\displaystyle=i\sigma^{2}\otimes\sigma^{k}$

satisfying the Clifford algebra

$\displaystyle\\{\gamma^{\mu},\gamma^{\nu}\\}$
$\displaystyle=2\eta^{\mu\nu}I_{4}\ .$

Taking the usual definition
$\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$ as the matrix which
anticommutes with all $\gamma^{\mu}$, let us write

$\displaystyle i\gamma^{5}\gamma^{0}$ $\displaystyle=\sigma^{2}\otimes I_{2}$
$\displaystyle-\gamma^{5}$ $\displaystyle=\sigma^{3}\otimes I_{2}$

which give

$\displaystyle(\vec{\sigma}\cdot\hat{q}\otimes I_{2})$
$\displaystyle=\frac{1}{|\vec{q}|}\gamma^{0}\gamma^{5}(\gamma^{5}q_{1}-iI_{4}q_{2}+\gamma^{0}q_{3})$
(21) $\displaystyle(I_{2}\otimes\vec{\sigma}\cdot\hat{p})$
$\displaystyle=-\frac{1}{|\vec{p}|}\gamma^{0}\gamma^{5}\vec{\gamma}\cdot\vec{p}\
.$ (22)

Rewriting the operators of Eqs. (15) and (16) in terms of the gamma matrices,
we find

$\displaystyle(\vec{\sigma}\cdot\hat{q}\oplus\vec{\sigma}\cdot\hat{p})=\frac{1}{|\vec{p}||\vec{q}|}\gamma^{0}\gamma^{5}(\gamma^{0}|\vec{p}|q_{3}-|\vec{q}|\vec{\gamma}\cdot\vec{p}+\gamma^{5}|\vec{p}|q_{1}-iI_{4}|\vec{p}|q_{2})$
$\displaystyle(\vec{\sigma}\cdot\hat{q}\ominus\vec{\sigma}\cdot\hat{p})=\frac{1}{|\vec{p}||\vec{q}|}\gamma^{0}\gamma^{5}(\gamma^{0}|\vec{p}|q_{3}+|\vec{q}|\vec{\gamma}\cdot\vec{p}+\gamma^{5}|\vec{p}|q_{1}-iI_{4}|\vec{p}|q_{2})\
.$

It is our goal to identify the vector $\vec{q}$ with our physical quantities.
Scaling these operators from the left with
$\pm|\vec{p}||\vec{q}|\gamma^{5}\gamma^{0}$, we now restrict $\vec{q}$ and
$\vec{p}$ to be of equal-magnitude (this is equivalent to the on-mass-shell
asssumption). We then obtain equivalent equations to Eqs. (15) and (16) of the
form

$\displaystyle(-\gamma^{0}q_{3}+\vec{\gamma}\cdot\vec{p}+\gamma^{5}q_{1}-iI_{4}q_{2})\chi_{\pm,\mp}$
$\displaystyle=0$ (23)
$\displaystyle(+\gamma^{0}q_{3}+\vec{\gamma}\cdot\vec{p}-\gamma^{5}q_{1}+iI_{4}q_{2})\chi_{\pm,\pm}$
$\displaystyle=0\ .$ (24)

It is now possible to make the identifications with our physical quantities
explicit. For the vector $\vec{q}$, we identify:

$q_{1}=im\sin{\alpha}\qquad q_{2}=-im\cos{\alpha}\qquad q_{3}=p_{0}=E$ (25)

The identification of $q_{3}=E$ ($E>0$) is chosen to align with our definition
of gamma matrices though equivalent linear combinations of the vector may be
found through unitary transformations. It may appear curious upon first
inspection that the vector $\vec{q}$ is seemingly compelled to take on
imaginary values in two components and real values in the third. It is,
however, a simple matter to absolve ourselves of this inhomogeneity by
performing a Wick-rotation of the energy axis in the complex plane and thereby
considering the four vectors of Minkowski spacetime in purely Euclidean terms.
In this way the connection between our projection operators as a basis of
$SL(2,\mathbb{C})$ and the Lorentz group $SO(1,3)$ may be made explicit. These
simplifications notwithstanding, we will continue to consider real-valued
energies in Minkowski spacetime.

Substituting our terms of Eqs. (25), into Eqs. (23) and (24) we obtain

$\displaystyle(-\gamma^{0}E+\gamma^{k}p_{k}+me^{i\alpha\gamma^{5}})\chi_{\pm,\mp}$
$\displaystyle=0$ (26)
$\displaystyle(+\gamma^{0}E+\gamma^{k}p_{k}-me^{i\alpha\gamma^{5}})\chi_{\pm,\pm}$
$\displaystyle=0\ .$ (27)

It is clear that Eqs. (26) and (27) are the momentum space analogues of the
CDE for positive and negative energies and therefore are equivalent to plane
wave solutions of Eq. (3). The eigenstates therefore satisfy

$\displaystyle E^{2}\chi_{s_{1},s_{2}}=(p^{2}+m^{2})\chi_{s_{1},s_{2}}\ .$
(28)

The method of projection operators for deriving fundamental equations has
potential to extend beyond the $\mathbb{C}^{4}$ vector space of bi-spinors.
While outside the scope of the present work, the possibility of extending the
concepts and methodologies presented here to investigate the algebraic
structure inherent in three particle flavors and their mass spectrum remains a
tantalizing possibility.

##  3 Symmetries and their physical implications

###  3.1 Spacetime symmetries

From the Special Theory of Relativity we know that in order for an equation to
be considered fundamental it must remain invariant in all inertial frames of
reference. Such frames may be defined as those frames in which the symmetries
of Minkowski spacetime are agreed to hold. These symmetries are represented by
the Poincaré group $\mathcal{P}\ =\ SO(3,1)\otimes_{s}T(3+1)$ consisting of
rotations, boosts, and translations in space and time (see Section 2.1) which
carry one inertial frame to another. We may therefore refer to the class of
inertial observers to whom fundamental equations must remain invariant as as
Poincaré observers. It follows as a neccesary condition for any equation to be
considered fundamental that it preserve its form for all Poincaré observers
and hence remain invariant with respect to all transformations given by
$\mathcal{P}$. Only such equations will be able to make physical predictions
about which all Poincaré observers will agree upon. It is easy to verify that
the derived CDE above is one such Poincaré invariant equation and so satisfies
a necessary condition to be called a fundamental equation of physics.

The other characteristics requisite of a fundamental theory may be summarized
as locality, gauge invariance, and the satisfaction of a least-action
principle (equivalent to the existence of Lagrangian density for the
equation). Since the CDE is a first-order partial differential equation, it is
local. In this paper only free particles are considered, therefore, gauge
invariance is not included. In Section 2.2, we demonstrated that the
Lagrangian density for the CDE exists and may be written in the fully
symmetric form given by Eq. (6). In general Lagrangians posses less symmetry
than the dynamical equations resulting from them due to the assumptions on
which the Noether theorem is based [18,19]. The best known example is the law
of inertia, whose dynamical equation is Galilean invariant but its Lagrangian
is not [20,21]. However, a method to restore Galilean invariance of the
Lagrangian was developed and applied to the law of inertia [22].

We followed [17] to construct the Dirac Lagrangian, which is already Poincaré
invariant. Similarly, the Lagrangian for the CDE equation is also Poincaré
invariant. Therefore the form of the CDE equation and its Lagrangian, as well
as theoretical predictions resulting from them, are the same for all observers
who accept the Principle of Relativity underlying Special Theory of Relativity
and its Poincaré group $\mathcal{P}$. Therefore, the presented results
combined with the above discussion clearly imply that any theory of physics
based on the CDE equation and its Lagrangian may be rightly called a
fundamental equation of physics.

###  3.2 CPT symmetries

Beyond the symmetry group of $\mathcal{P}$, we may also inquire into which
discrete symmetries of nature hold for the CDE. We define the operations of
charge-conjugation ($\mathcal{C}$), spatial inversion ($\mathcal{P}$), and
time-reversal ($\mathcal{T}$) on an arbitrary operator $\hat{O}(x,t)$ acting
on a state $\psi(x,t)$ as follows

$\displaystyle\mathcal{C}[\hat{O}(x,t)\psi(x,t)]\mathcal{C}^{-1}$
$\displaystyle=\hat{O}^{*}(x,t)\psi^{*}(x,t)$ (29)
$\displaystyle\mathcal{P}[\hat{O}(x,t)\psi(x,t)]\mathcal{P}^{-1}$
$\displaystyle=\hat{O}(-x,t)\psi(-x,t)$ (30)
$\displaystyle\mathcal{T}[\hat{O}(x,t)\psi(x,t)]\mathcal{T}^{-1}$
$\displaystyle=\hat{O}^{*}(x,-t)\psi(x,-t)\ .$ (31)

With these definitions, we may determine the parity-conjugated forms of the
CDE. Defining
$\hat{D}(\alpha)=i\gamma^{\mu}\partial_{\mu}-me^{i\alpha\gamma}$, taking the
symmetry operation $\mathcal{U}\in\\{\mathcal{C},\mathcal{P},\mathcal{T}\\}$,
and writing the corresponding unitary transformation acting on the spinor
indices $\hat{U}\in\\{\hat{C},\hat{P},\hat{T}\\}$, we obtain

$\displaystyle\hat{U}[\mathcal{U}\hat{D}(\alpha)\mathcal{U}^{-1}]\hat{U}^{\dagger}\psi_{U}(x,t)$
$\displaystyle=0$ (32)

where $\psi_{U}(x,t)\equiv\hat{U}[\mathcal{U}\psi(x,t)\mathcal{U}^{-1}]$.
Then, the resulting transformations may be summarized as

$\displaystyle\hat{C}[\mathcal{C}\hat{D}(\alpha)\mathcal{C}^{-1}]\hat{C}^{\dagger}$
$\displaystyle=\hat{D}(\alpha^{*})$ (33)
$\displaystyle\hat{P}[\mathcal{P}\hat{D}(\alpha)\mathcal{P}^{-1}]\hat{P}^{\dagger}$
$\displaystyle=\hat{D}(-\alpha)$ (34)
$\displaystyle\hat{T}[\mathcal{T}\hat{D}(\alpha)\mathcal{T}^{-1}]\hat{T}^{\dagger}$
$\displaystyle=\hat{D}(-\alpha^{*})\ ,$ (35)

and the following conditions can be identified

1\. If $\psi$ exhibits C-invariance ($\psi=\psi_{C}$), then
$\alpha\in\mathbb{R}$

2\. If $\psi$ exhibits CP-invariance ($\psi=\psi_{CP}$), then
$\alpha\in\mathbb{I}$

3\. If $\psi$ exhibits CPT-invariance ($\psi=\psi_{CPT}$), then
$\alpha\in\mathbb{C}$

These conditions combined with the CPT theorem reinforce our conclusion that
the Dirac equation with chiral freedom is the most general first order
differential equation derivable from the irreps of the Poincaré group. While
it is quite conceivable that the constraints imposed by observed classes of
discrete symmetries in interactions (and their fundamental violations) may
contribute to constraints on the CDE, it is important to emphasize the regimes
of validity presented here are derived for free particles in the absence of
interactions and therefore a result of extrinsic and not intrinsic symmetries.
In this way the discrete constraints presented above are foundational for any
extended theoretical considerations.

##  4 Conclusions

We have presented three distinct derivations of the generalized form of
Dirac’s equation to include all those parameters not constrained by Poincaré
symmetry: from the irreducible representations of the Poinccaré group, from
Lagrangian formalism, and from a classification scheme built from projection
operators. The result of all these derivations agree and show the chiral angle
is a necessary degree of freedom for complete categorization of all physical
eigenstates of spin and the matter/anti-matter nature. Equivalently, by
showing that the obtained chiral Dirac equation is local, Poincaré invariant
and its Lagrangian exists, the equation becomes a fundamental equation of
modern physics. Using this derived equation it has then possible to
investigate the underlying spacetime and CPT symmetries and the constraints on
allowable values.

Author Contributions: Conceptualization, T.B.W. and Z.E.M..; methodology,
T.B.W. and Z.E.M.; validation, T.B.W. and Z.E.M.; formal analysis, T.B.W.;
investigation, T.B.W. and Z.E.M.; writing—original draft preparation, T.B.W.
and Z.E.M.; writing—review and editing, T.B.W. and Z.E.M. Both authors have
read and agreed to the published version of the manuscript.

Funding: This research received no external funding.

Instututional Review Board Statement: Not applicable.

Informal Consent Statement: Not applicable.

Data Availability Statement: Not applicable.

Acknowledgments We are grateful to two reviewers for reading our paper and
providing comments and suggestions for its improvement. Our special thanks to
M. Efroimsky (one of the reviewers) for bringing to our attention differences
between our derivation and that presented by Ryder [15].

Conflict of Interest: The authors declare no conflict of interest.

## References

  * [1] P.A.M. Dirac, The quantum theory of the electron, Proc. Royal Soc. London 117 (1928) 610
  * [2] I. Sogami, Prog. Theor. Phys. 66 (1981) 303
  * [3] S.I. Kruglov, arXiv:hep-ph / 0507027v2 23 June 2006
  * [4] E. Marsh, Y. Narita, Front. Phys. 3 (2015) Article 82
  * [5] A.O. Barut, P. Cordero, G.C. Ghirardi, Phys. Rev. 182 (1969) 1844
  * [6] A.O. Barut, The mass of muon, Phys. Lett. 73B (1978) 310
  * [7] W. Pfister, Mixed-symmetry solutions of generalized three-particle Bargmann-Wigner equations in the strong-coupling limit, Nuovo Cimento A 108 (1995) 1427
  * [8] S.I. Kruglov, On the Hamiltonian form of generalized Dirac equation for fermions with two mass states, Elect. J. Theor. Phys. 3 (2006) 11
  * [9] S.I. Kruglov, Modified Dirac equation with Lorentz invariance violoation and its solutions for particles in an external magnetic field, Phys. Let. B 718 (2012) 228
  * [10] D. Leiter, G. Szamosi, Pseudoscalar mass and its relationship to conventional scalar mass in relativistic Dirac theory of the electron, Let. Nuovo Cim. 5 (1972) 814
  * [11] K. Nozari, Generalized Dirac equation and its symmetries, Chaos, Solitons & Fractals 32 (2007) 302
  * [12] T.B. Watson, Z.E. Musielak, Chiral symmetry in Dirac equation and its effecst on neutrino masses and dark matter, Int. J. Mod. Phys. A 35 (2020) 2050189
  * [13] E.P. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. Math. 40 (1939) 149
  * [14] Y.S. Kim and M.E. Noz, Theory and Applications of the Poincaré Group, Reidel, Dordrecht, 1986
  * [15] L.W. Ryder, Quantum Field Theory, Cambridge University Press, Cambridge, 1985
  * [16] P.H. Frampton, Gauge Field Theories, John Wiley & Sons, Inc., New York, 2000
  * [17] N.A. Daughty, Lagrangian Interactions, Addison-Wesley Publ. Comp., Inc., Sydney, 1990
  * [18] S. Hojman, Symmetries of Lagrangians and of their equations of motion, J. Phys. A: Math. Gen. 17 (1984) 2399
  * [19] S. Hojman, A new conservation law constructed without using either Lagrangians or Hamiltonians, J. Phys. A: Math. Gen. 27 (1992) L59
  * [20] L.D. Landau and E.M. Lifschitz, Mechanics, Pergamon Press, Oxford, 1969
  * [21] J.-M. Levy-Leblond, Group-theoretical foundations of classical mechanics: the Lagrange gauge problem, Comm. Math. Phys. 12 (1969) 64
  * [22] Z. E. Musielak and T. B. Watson, gauge functions and Galilean invariance of Lagrangians, Phys. Let. A 384 (2020) 126642
  * [23] M. Petitjean, About chirality in Minkowski spacetime, Symmetry, 11 (2019) 1320