∎

11institutetext: CVIT, International Institute of Information Technology,
Hyderabad, India
email<EMAIL_ADDRESS>
# Camouflaged Object Detection and Tracking: A Survey

Ajoy Mondal

###### Abstract

Moving object detection and tracking have various applications, including
surveillance, anomaly detection, vehicle navigation, etc. The literature on
object detection and tracking is rich enough, and several essential survey
papers exist. However, the research on camouflage object detection and
tracking is limited due to its complexity. Existing work on this problem has
been done based on either biological characteristics of the camouflaged
objects or computer vision techniques. This article reviews the existing
camouflaged object detection and tracking techniques using computer vision
algorithms from the theoretical perspective. This article also addresses
several issues of interest as well as future research direction in this area.
We hope this review will help the reader learn the recent advances in
camouflaged object detection and tracking.

###### Keywords:

Camouflaged object, detection, and tracking.

##  1 Introduction

Object detection is a technique which deals with detecting instances of
semantic objects of a specific class (such as human, buildings, cars, etc.) in
digital images and videos. It has several computer vision applications,
including image retrieval, video surveillance, and some other image and video
analysis tasks. The considerable success is achieved for object detection
problems in a controlled environment, but the issue remains unsolved in wild
places.

a b

c d

e f

g h

i

Figure 1: A visual illustration of various challenges in object tracking.
Examples of (a) illumination variation, (b) background clutter, (c) partial
occlusion, (d) full occlusion, (e) change in object’s scale, (f) change in
object’s orientation, (g) camouflaged object, (h) pose variation, and (i)
irregular shaped object.

Visual object detection and tracking become a very challenging problem due to
several factors like (i) low-quality camera sensors (including low resolution,
low bit depth, low frame rate and color distortion), (ii) challenging factors
(like tracking non-rigid object, tracking small object, tracking multiple
objects and tracking pose varying object), (iii) requirements for real-time
tracking, (iv) multi-view object tracking and (v) variations in object
appearance due to several complicated factors (such as illumination variation
(Figure 1(a)), background clutter (Figure 1(b)), partial object occlusion
(Figure 1(c)), full object occlusion (Figure 1(d)), large variation in object
scale and orientation (Figures 1(e) and (f)), partially camouflaged objects
(Figure 1 (g)), pose variation (Figure 1 (h)), shape deformation (Figure 1
(i)), rapid camera motion and noise yilmaz2006object ; smeulders2013visual ;
li2013survey . Detection and (or) tracking accuracy may be degraded and even
failed due to these challenges. Numerous object tracking algorithms have been
developed in the literature to handle these challenges. These invented
algorithms with different properties and characteristics usually solve
different visual object detection and tracking problems.

Sometimes, objects hide their signatures into their surroundings and create
camouflage. The occurrence of camouflage makes object detection a more complex
problem. According to Copeland and Trivedi copeland1997models , “camouflage is
an attempt to conceal the signature of a target into the background”. In other
words, camouflage is the ability of prey to hide from predators by changing
their body pattern, texture, and coloration as per the environment’s texture.
A camouflaged object cannot be adequately visible by human vision systems. In
this context, computer vision-based approaches are proposed to analyze the
camouflaged objects. Work-related to camouflage can be roughly divided into
two major areas: (i) camouflage assessment and design, (ii) camouflage
detection, or breakin copeland1997models ; gretzmacher1998camouflage ;
zhangcamou . A camouflage detection system or de-camouflaging is used to
extract a target from its background. It discriminates foreground object from
camouflaged image frames singh2013survey ; rong2016camouflage . Camouflage
detection or breaking system has many potential applications, including (i)
preserving wildlife, (ii) enemy detection in the battlefield, (iii) defect
detection during manufacturing, (iv) identification of duplicate products
during logistics, etc singh2013survey . Some animals have unique biological
characteristics to make them camouflaged in the environment. More research
work has been done for camouflage breaking based on the biological property of
these animals. The vision features of a camouflaged object are very similar to
the background. The color of a camouflaged object is the same as the
surrounding environment, and the texture is destroyed to merge with the
background. Such characteristics of camouflaged objects make detection and
tracking tasks more difficult. Due to the complexity of the problem, less work
has been done using computer vision-based techniques. In this article, we
review existing computer vision-based approaches for detection and tracking of
a camouflaged object. We also discuss the merits and demerits of each of the
algorithms. We also discuss various issues of the existing algorithms and
future direction on this particular topic. We hope this review will help the
reader learn the recent advances in camouflaged object detection and tracking.

The rest of the article organizes as follows. Section 2 discuss different
existing detection and tracking algorithms for camouflaged object. Conclusive
remark and future direction are presented in Section 3.

##  2 Camouflaged Objects Detection and Tracking

Visual features of a camouflaged object are very much similar to the
background — (i) the intensity or color of the camouflaged object is close to
their surrounding environment, (ii) the texture is destroyed to merge with the
background, and (iii) the boundary of the camouflaged object is blurred. Such
visual characteristics of camouflaged objects make detection and tracking
tasks more difficult. Due to such complexity, less work has been done to
attempt visual camouflage breaking in literature. However, the researchers
have developed various algorithms using various visual features (e.g.,
intensity or color, texture, motion, gradient, etc.) to detect camouflaged
objects from their surroundings. Here, we try to group the existing methods
according to visual feature considering for detection and tracking of
camouflaged objects. In the following subsection, the current techniques on
consideration of each of the visual features are discussed.

###  2.1 Intensity/Color Features

The feature plays an essential role in the detection of camouflaged objects.
Here, techniques developed based on the intensity/color values of the frames
are mainly discussed. Boult _et al._ boult2001into  developed a background
subtraction technique with two thresholds to detect the camouflaged target.
Here, a higher threshold value is used to detect pixels that are certainly in
the foreground. The lower one is considered to detect uncertain pixels (i.e.,
pixels that are either part of the background or camouflaged part of the
object). Then, the quasi connected component is taken into consideration to
get the camouflaged target. In this case, detection accuracy is also highly
dependent on thresholds. The selection of proper threshold value itself is a
problem. For slow-moving objects, this method fails to detect objects.

On the other hand, Hung and Jiang huang2005tracking  devised a method to track
a camouflaged object using sequential execution of weighted region
consolidation and active contour. An iterative weighted region consolidation
operator is used to fill the gaps introduced by camouflage. Then, an active
contour model is built during tracking to capture the actual shape of the
target. The performance of this method relies on the inter-frame difference.
If the object has slow motion, it is challenging to localize the object using
an iterative weighted region consolidation operator. Hence, tracking may fail
for sequences containing slow-moving and uniform colored objects.

In boot2009training  Boot et al. discussed that we usually learn something
general about target recognition, which allow us to guide our eyes to the
target more efficiently and recognize it faster and farther from fixation.
They described that the background contains regular patterns. Deviations from
this regularity signify the presence of a camouflaged target. However, this
does not happen for all kinds of camouflaged objects. If the background and
object contain similar regular patterns, it becomes challenging to extract the
object from its surrounding. To detect a camouflaged object, Chandesa _et al._
chandesa2009detecting  proposed an algorithm based on particle filter. Here,
the Gaussian mixture model of particle distribution is considered to
investigate camouflage’s effect on the particle set representing the object.
This method works well on the occluded object but not for the camouflaged
object. Though this method works well, it needs object information (a priori)
to execute the algorithm. In conte2009algorithm , Conte _et al._ proposed an
algorithm to detect partially camouflaged people. Here, background subtraction
is used to detect different parts of a person. Then grouping is performed
based on a model of the shape of targets. This method is unable to provide
satisfactory results for objects other than humans. In zhang2016bayesian , a
camouflaged model is proposed using a global model for the background and
integration of global and local models for the foreground. Here, both the
models helped to detect camouflaged objects.

#### Discussion:

In general, intensity or color features are elementary and computationally
efficient for fast, camouflaged object detection and tracking. The intensity
or color feature can detect camouflaged objects where camouflage is occurred
due to texture similarity with the background. In contrast, these features
cannot detect camouflaged objects where camouflage occurs due to color
similarity with the environment.

###  2.2 Motion Features

Motion is considered as an essential feature to detect an object. Several
techniques have been developed based on the motion of the objects. McKee _et
al._ mckee1997stereopsis  concluded from their experiments that stereopsis is
generally useful on breaking camouflage when both the observer and the scene
are non-dynamic. Here, motion is a helpful feature for breaking camouflage on
a static background. If the background is non-static, the motion feature fails
to extract camouflaged objects from its surrounding. In this direction,
Ternovskiy and Jannson ternovskiy_1997  have proposed a motion prediction
approach to detect the target in the camouflage environment. This method is
suitable for the sequences where changes occur due to the object and camera
movement only. If changes occur due to illumination variation, this method
considers changes occurring due to object movement. Hence, this method can not
work correctly in such a situation. On the other hand, for breaking
camouflage, Huimin _et al._ huimin1999  developed a computational model of
visual moving image filtering in which Reichardt’s elementary motion detectors
reichardt1989processing  are employed for detecting motion information. As
this method relies on motion (due to object movement) information, motion due
to other conditions like illumination variation, environmental condition
changes, etc. produces more false alarms. All these techniques mentioned above
are context-dependent and may not work well for various types of camouflaged
objects.

In hou2011detection , Yin _et al._ developed an algorithm to track a mobile
object with a camouflage color based on the optical flow model. Here, the
optical flow model is used to detect motion patterns of the object and the
background. The motion patterns are clustered and detect the camouflaged
object based on the optical flow’s magnitude and location. After that, the
Kalman filter is used to improve the detection accuracy. However, the accuracy
of this model depends on the results of the optical flow. For slow-moving
objects and objects with camera motion, this method fails to provide excellent
results.

#### Discussion:

motion plays a vital role in detecting camouflaged objects in the literature.
Motion features help to detect camouflaged objects while camouflage occurs due
to color/texture similarity with the background. However, motion features also
fail to detect a camouflaged object with prolonged movement or stop/go motion.

###  2.3 Texture Features

Sometimes, the object’s color is similar to the background, but they have
different texture patterns. The texture is considered to discriminate against
the object from its surrounding. Galun _et al._ galun2003texture  developed a
technique to detect camouflaged objects using a bottom-up aggregation
framework that combines structural characteristics of texture elements with
filter responses. It adaptively identifies the shape of texture elements and
characterizes them by their size, aspect ratio, orientation, brightness, etc.
Then various statistical measures of these properties are taken into account
to distinguish between different textures. The said approach is applied to
images containing various kinds of textures. This method works well for images
containing different textures for objects and backgrounds. However, if the
object and background contain a similar texture, this technique may fail to
produce good results. In bhajantri2006camouflage , Nagabhushan and Bhajantri
proposed a technique for multiple camouflage breaking using co-occurrence
matrix robert1973textural  and Canny edge detector canny1986computational .
The co-occurrence matrix is used to analyze the given image’s texture, whereas
the Canny edge detector is considered to detect the edges. A combination of
both co-occurrence matrix and the Canny edge detector enhances the
separability between objects containing different textures. Though this method
provides good results for synthetic images, it is not applied to real-life
data. Also, background information needs to be known before executing this
method.

Neider and Zelinsky discussed in neider2006searching  the detection of
camouflaged targets by looking through the distracters or by scrutinizing the
target-similar background. In bhajantri2006camouflage , Bhajantri and
Nagabhushan proposed a technique to detect the camouflaged defect. Here, co-
occurrence matrix-based texture features are computed within a small image
region. The defective portion is detected by cluster analysis and watershed
segmentation. The accuracy of this method depends on the texture feature. It
may not work well for sequences where objects and background contain similar
kind of texture. Sengottuvelan _et al._ sengottuvelan2008performance
developed a technique to detect the camouflaged portion of the object and
extract it from the environment in a given image. Here, the grey level co-
occurrence matrix (glcm) based texture feature and dendrogram are used to
detect the camouflaged object. This technique is very time-consuming due to
the given image’s division into several blocks or smaller regions. It does not
work for images containing shading effects and object & background containing
similar textures.

Liming and Weidong liming2010new  proposed a technique based on weighted
structural similarity (wssim) to design and evaluate camouflage texture. Here
they used weighted structural similarity and original image feature to create
a camouflage image. It can be used for breaking the camouflage. In
owens2014camouflaging , Owens et al. introduced several background matching
algorithms that attempt to make the object look like whatever is behind it. It
is impossible to match the background from every possible viewpoint exactly.
But the proposed models are forced to make trade-offs between different
perceptual factors, such as conspicuousness of occlusion boundaries and the
amount of texture distortion.

In the same direction, Li et al. proposed a texture guided weighted voting
(tgwv) method detect foreground object in camouflaged scenes li2017foreground
. This method employed the stationary wavelet transform to decompose the image
into frequency bands. This technique could effectively capture small and
hardly noticeable differences between the foreground and background in the
image domain in certain wavelet frequency bands. Finally, the foreground is
detected using a weighted voting scheme based on all the wavelet bands’
intensity and texture. Experimental results demonstrate that this method
achieves superior performance compared to the current state-of-the-art
results.

#### Discussion:

Though texture feature extraction from the color or intensity is costly.
However, it is more effective for detecting camouflaged objects. While
camouflage occurs due to color similarity with background, texture features
give promising results in such cases.

###  2.4 Gradient Features

When the object has a similar color, texture as the background, it is
challenging to detect objects using these features. For those sequences,
gradient information is useful to extract the object from the background
region. In this direction, various methods have been developed to detect the
camouflaged object. In tankus1998detection ; tankus2000convexity ;
tankus2000model ; tankus2001convexity ; tankus2009computer ; pan2011study ,
Tankus and Yeshurum proposed a $D_{arg}$ operator (context-free) to enhance an
area whose shading corresponds to a convex (or concave) $3D$ object to
separate such area from a flat background having similar features (like color
and texture). $D_{arg}$ is applied directly to the grey-level function of the
image. It responds to smooth three-dimensional convex or concave patches in
objects and is not limited by any particular light source or reflectance
function. Results obtained using the $D_{arg}$ operator are highly dependent
on threshold values. The selection of a suitable threshold is a major issue.
Further, this method does not work well for an environment containing concave
background and dark-colored objects.

#### Discussion:

Gradient features are useful in case of camouflage occur due to
intensity/color and texture similarity with the background.

###  2.5 Combination of Various Features

Camouflage occurs when the visual characteristics of the objects are too
similar to the background. The objective of a camouflage detection system is
to separate the camouflaged object from the background. In such cases,
features generated from single-cue (like color or texture or shape or motion)
is not sufficient to extract camouflaged objects from the background because
the underlying phenomenon for the occurrence of camouflage is not known. In
this context, the integration of features generated from multi-cue (color,
texture, shape, motion, etc.) may increase the separability between the
camouflaged objects and background. Various algorithms have been proposed
based on integrating different features like color, texture, motion, shape,
etc.

In this direction, to detect camouflaged objects, Harville _et al._
harville2001foreground  proposed a foreground segmentation technique using
both color and depth information. Information loss occurs due to a $3$D
scene’s projection into a $2$D picture. Here, depth information is considered
to reduce the information loss due to $3$D to $2$D projection. The use of
depth information increases detection accuracy and also increases the
computational cost. In kaewtrakulpong2003real , Pong and Bowdenb proposed the
use of the stochastic process to handle camouflage. This algorithm assumes
that camouflage occurs when the new observation can not be associated with an
existing task. The performance of this method depends on the color and motion
of the object. Here, color is used to distinguish the camouflaged object and
its surrounding, whereas motion is considered to separate out moving objects
and the static part of the frame. Hence, both this information helps to detect
moving camouflaged objects. Sometimes, this method may fail for an object,
which is very similar to the background, and it contains more static objects.
In brady2003bootstrapped , Brady and Kersten proposed a computational approach
to combine low-level features with high-level models to detect and recognize
an object when camouflage is present. Here, bootstrapped learning is
considered to build the shape of the object, and it depends on the result of
object detection. If object detection is not perfect, then recognition may not
always be correct, and it fails to track correctly.

Furthermore, in loza2009structural , Losa et al. proposed a particle filter
based on structural similarity measure for tracking camouflaged objects. Here,
the structural similarity measure reflects the distance between two frames by
comparing their luminance, contrast, and spatial characteristics. Here, the
used measure is sensitive to relative rather than absolute changes in the
frame. However, this technique may not be robust to significant alteration of
the tracked object.

However, Jiang jiang2011object  proposed a model to track the object in a
video sequence by locating a list of object features that are ranked according
to their ability to differentiate against the background. Here, a mixture of
color, texture, and object motion is considered multidimensional to represent
the object. Multi-features representation increases the separability between
the objects and their surrounding. Experimental results show that this model
works well for detecting the camouflaged object.

Recently, Mondal et al. developed a camouflaged object tracking algorithm
using a probabilistic neural network and fuzzy energy-based active contour
mondal2016partially . Here, multi-features like color, texture, and shape are
integrated to increase the discrimination between camouflaged objects and
their background. The camouflaged object is detected by modifying the
probabilistic neural network, and finally, it is tracked by the proposed fuzzy
energy-based active contour model. The experiments showed that this method
could provide good tracking results under a camouflaged environment.

#### Discussion:

Color features work while camouflage occurs due to texture similarity with the
background. Texture features are effective, while camouflage occurs due to
color similarity. The motion of the camouflaged object can detect it while the
object with good motion. These features, e.g., color, texture, and motion, can
detect the camouflaged object in a specific environment. A combination of
various features, e.g., color, texture, motion, etc. effectively work without
constraints.

###  2.6 Deep Features

As the color of the camouflaged object is similar to the background, detection
of it is challenging. In li2017camouflage , the authors considered the image
enhancement technique to increase the discrimination between object and
background. The region proposal network (rpn) is considered to estimate the
required target’s accurate positioning. The deep neural network is used to
identify extracted RoI. Finally, the detection of a camouflaged object is
done. In this direction, Fang et al. deep_2019  proposed strong semantic
dilation network to detect camouflaged people. The authors claimed that full
use of semantic information in convolutional neural networks and dilated
convolutions are also added to enlarge the receptive field to find camouflage
people. The authors presented impressive results on a standard dataset.

#### Discussion:

All the features, e.g., color, texture, motion, and gradient, are hand-crafted
and may not use for all types of camouflaged objects. On the contrary, the in-
depth feature is learned by the network from the extensive training images.
The deep feature is more generic than the hand-crafted features. It works
better than hand-crafted features.

###  2.7 Other Techniques

Some other approaches that exist in the literature are discussed in this
section. Marouani _et al._ marouani1995model  developed aircraft recognition
technique in the presence of camouflage. It works on low-level matching
between segments of the projection of a $3D$ model of the objects of interest.
In Guilan_1997 , Guilan and Shunqing discussed the use of spectral pattern
recognition techniques to distinguish color camouflage from green vegetation
background. Here, the considered spectral feature is the basic characteristic
for recognizing the target. Experimentally, they concluded that the considered
feature is optimum for recognition than other state-of-the-art features.
However, it cannot provide good results for camouflaged objects in the
background other than green vegetation.

Beiderman et al. beiderman2010optical  presented a novel approach where the
secondary speckle pattern is monitored for a time, to extract the
temporal/spectral signature of the objects. Special image processing
algorithms allow obtaining a unique signature of the object region that can be
used to classify, recognize, and identify objects. This approach can detect
and recognize camouflaged objects.

Pan et al. pan2011study  developed an algorithm to detect the camouflaged
portion of the object in a complex background with the help of $D_{arg}$
operator. The main drawback of this method is the selection of proper
threshold value. Liu and Huang liu2012foreground  developed a novel foreground
object detection scheme that integrates the top-down information based on the
expectation-maximization (em) framework. Here, top-down information is
incorporated into the object model in a generalized em framework. A foreground
model is constructed based on the object model and the state of each target.
The author concluded that the method is giving good results for detecting
camouflaged objects. Recently, Malathi and Bhuyan joy2013department  developed
a background subtraction scheme to detect the camouflaged object. Here, pixels
corresponding to the background are quantized into codebooks, sufficient to
represent a background. Codebooks extracted for each of the cameras are
combined to extract the foreground from the background. To detect the
camouflaged target in complex background, Zhou and Cun-chao zhou2013cluster
proposed a novel spectral-polarimetric image fusion algorithm based on
Shearlet transform. Kernel fuzzy c-means clustering algorithm is applied to a
fused spectral-polarimetric image to separate camouflaged object from its
background.

In kim2014unsupervised , Kim proposed a fully autonomous feature selection and
camouflaged object detection method based on the online analysis of spectral
and spatial features for hyper-spectral images. Here, a statistical distance
metric is considered to generate candidate feature bands, and entropy-based
spatial grouping property is used to reduce useless feature bands. Camouflaged
objects are detected by optical spectral-spatial feature analysis with less
computational complexity. Mangale and Khambete mangale2016camouflaged  fused
Thermal infrared and visible spectrum imaging modality to detect camouflaged
objects. The authors presented impressive results. Li et al. li2018fusion
proposed a camouflaged object detection model in wavelet transformed feature
space. In this method, the likelihood of each wavelet coefficient being
foreground is estimated by formulating foreground and background models for
each wavelet band. This method effectively aggregates the likelihoods from
different wavelet bands based on the wavelet transforms’ characteristics
detects camouflaged objects.

##  3 Conclusion and Future Directions

In this article, a review of camouflaged object detection and tracking using
computer vision-based approaches are presented. Due to the complexity of the
problem, less research work has been done for breaking camouflage. Among them,
very few works have been done for extracting camouflaged objects in an
unsupervised way. It is crucial to detect the camouflaged object in the real
scenario. As color, texture, and shape of the camouflaged object are very
similar to its surroundings, techniques based on the single feature cannot
extract camouflaged objects well. The integration of several features like
color, texture, motion, shape, and some other features related to camouflaged
objects’ biological characteristics may be considered to improve the
camouflage braking system. However, camouflage breaking has several crucial
real-life applications, including soldier detection on the battlefield and
wildlife preservation. In such cases, the unsupervised technique is necessary
to detect such camouflaged objects.

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# One-Sided $k$-Orthogonal Matrices
Over Finite Semi-Local Rings
And Their Codes

Virgilio P. Sison
Institute of Mathematical Sciences and Physics
University of the Philippines Los Baños
College 4031, Laguna, Philippines
Email<EMAIL_ADDRESS>

Charles R. Repizo
Department of Mathematics and Physics
University of Santo Tomas
España Blvd., Sampaloc, Manila 1008, Philippines
Email<EMAIL_ADDRESS>
###### Abstract

Let $R$ be a finite commutative ring with unity $1_{R}$ and $k\in R$.
Properties of one-sided $k$-orthogonal $n\times n$ matrices over $R$ are
presented. When $k$ is idempotent, these matrices form a semigroup structure.
Consequently new families of matrix semigroups over certain finite semi-local
rings are constructed. When $k=1_{R}$, the classical orthogonal group of
degree $n$ is obtained. It is proved that, if $R$ is a semi-local ring, then
these semigroups are isomorphic to a finite product of $k$-orthogonal
semigroups over fields. Finally, the antiorthogonal and self-orthogonal
matrices that give rise to leading-systematic self-dual or weakly self-dual
linear codes are discussed.

Keywords: orthogonal matrices, semi-local rings, self-dual codes. Mathematics
Subject Classification (2020): 15B10, 13M99, 94B60.

##  1 Introduction

This study extends the classical theory of orthogonal matrices over fields to
the so-called one-sided (left or right) $k$-orthogonal matrices over rings,
where $k$ is some fixed scalar element. This new class of one-sided
$k$-orthogonal matrices is examined in the context of finite commutative rings
with unity. In particular, we focus our attention to finite semi-local rings
with at least two maximal ideals in order to fully utilize the Chinese
Remainder Theorem. Various properties of $k$-orthogonal matrices are described
and applications of these matrices in constructing linear codes are explored.
It is important to note at this point that we have given a different
definition of $k$-orthogonal matrices which may vary from those found in other
literature.

The material is organized as follows: Section 2 provides the important
conceptual framework while Section 3 contains the main results. MAGMA®
routines are designed to generate new examples of one-sided and two-sided
$k$-orthogonal matrices and to construct the codes from these matrices. A
simple algorithm to generate the set of idempotent elements of an arbitrary
finite ring is also written. Section 4 gives suggestions for further research.

##  2 Preliminaries and definitions

This section tackles briefly the concept of orthogonal matrices and the
general linear group. The classical theory of orthogonal matrices over the
finite field $\mathbb{F}_{q}$ is fully discussed by MacWilliams [1]. Some
important ring-theoretic concepts, such as coprime and maximal ideals,
idempotents, local rings, semi-local rings and finite chain rings, are also
introduced here. For a more thorough treatment of these topics, the reader is
referred to [2, 3, 4].

###  2.1 Ring of matrices and the orthogonal group

Let $R$ be a commutative ring with unity $1_{R}\neq 0$ and $M_{n}(R)$ the
unital noncommutative ring of square $n\times n$ matrices over $R$. A matrix
$A\in M_{n}(R)$ is nonsingular or invertible if and only if $\det(A)$ is a
unit in $R$, or equivalently, there exists $B\in M_{n}(R)$ such that
$AB=BA=I_{n}$, where $I_{n}$ is the identity matrix. The set of invertible
matrices in $M_{n}(R)$ forms a noncommutative multiplicative group and is
denoted by $GL_{n}(R)$. A matrix $A$ is symmetric if $A=A^{T}$, where $A^{T}$
is the transpose of $A$. It is called orthogonal provided the following holds.

$A^{T}A=AA^{T}=I_{n}.$ (1)

In other words, an orthogonal matrix is an invertible matrix whose unique two-
sided inverse is its transpose. In fact the set of orthogonal matrices over
$R$, denoted $O_{n}(R)$, is a (not necessarily normal) subgroup of
$GL_{n}(R)$, and is called the orthogonal group of degree $n$. It should be
remarked that, if $A\in O_{n}(R)$, then $\det(A)$ is either the identity or an
involution, but not conversely.

Let $A=[a_{ij}]$, $B=[b_{ij}]\in M_{n}(R)$, and $I$ an ideal of $R$. Then $A$
is said to be congruent to $B$ modulo $I$, written as $A\equiv_{n}B\pmod{I}$,
if and only if $a_{ij}\equiv b_{ij}\pmod{I}$, that is $a_{ij}-b_{ij}\in I$,
for all $i,j=1,2,...,n$. This is simply component-wise application of the
usual additive subgroup relation on an ideal. It is shown in [2] that
$\equiv_{n}$ is, quite expectedly, a congruence relation in $M_{n}(R)$.

Two ideals $I$ and $J$ of $R$ are said to be coprime if $I+J=R$. A set of
nontrivial ideals $\mathcal{I}=\\{I_{1},I_{2},...,I_{m}\\}$ in $R$ is pairwise
coprime if $I_{j}+I_{k}=R$ for all $j,k=1,2,...,m$ with $j\neq k$. Note that,
if $\mathcal{I}$ consists of distinct maximal ideals, then necessarily
$\mathcal{I}$ is pairwise coprime. The Chinese Remainder Theorem gives the
following powerful result for matrix rings.

######  Theorem 2.1 (Han, 2006)

Let $m$ and $n$ be any positive integers and $I_{1},$ $I_{2},$…,$I_{m}$ be
ideals in a ring $R$.Then there is a monomorphism of rings

$\theta:M_{n}\bigl{(}R/I_{1}\cap I_{2}\cap...\cap I_{m}\bigr{)}\rightarrow
M_{n}(R/I_{1})\times M_{n}(R/I_{2})\times\ldots\times M_{n}(R/I_{m})$

If $I_{1},I_{2},\dots,I_{m}$ are pairwise coprime, then $\theta$ is an
isomorphism.

This result extends naturally to the general linear group $GL_{n}(R)$, and
when $R$ is finite and the ideals $I_{j}$ are maximal, it gives a convenient
formula for $|GL_{n}(R)|$ as a product of the $m$ factors
$|GL_{n}(\mathbb{F}_{q_{j}})|$, in terms of the residue fields
$\mathbb{F}_{q_{j}}$, given by

$|GL_{n}(\mathbb{F}_{q_{j}})|=q_{j}^{\tfrac{n(n-1)}{2}}\displaystyle\prod_{i=1}^{n}(q_{j}^{i}-1),j=1,2,\ldots,m$
(2)

###  2.2 Finite semi-local rings

A semi-local ring is a commutative ring $R$ with unity with a finite number of
maximal ideals. In particular, if $R$ has a unique maximal ideal, then $R$ is
said to be a local ring. Hence a local ring is semi-local but not conversely.
The quaternary ring $\mathbb{F}_{2}+v\mathbb{F}_{2}$, where $v^{2}=v$, is a
semi-local ring with two maximal coprime ideals $(v)$ and $(1+v)$. The ring
$\mathbb{Z}_{6}$ of integers modulo $6$ is also a semi-local ring with two
maximal coprime ideals $(2)$ and $(3)$. A finite commutative ring with unity
whose ideals are linearly ordered by inclusion is a finite chain ring. The
ring $\mathbb{Z}_{p^{r}}$ of integers modulo $p^{r}$, $p$ prime and $r\geq 1$
an integer, is a local ring with unique maximal ideal $(p)$, and is a finite
chain ring of length $r$.

An idempotent of a ring is an element $a$ with the property that $a^{2}=a$,
that is, its square is itself. A Boolean ring is a ring in which all its
elements are idempotent. The ring $\mathbb{F}_{2}+v\mathbb{F}_{2}$, where
$v^{2}=v$, is an example of a Boolean ring. It is also known that, if
$a_{1},a_{2},..,a_{m}$ are idempotent elements of rings
$R_{1},R_{2},...,R_{m}$, respectively, then the $m$-tuple
$(a_{1},a_{2},...,a_{m})$ is idempotent in the cross product $R_{1}\times
R_{2}\times...\times R_{m}$. The integer ring $\mathbb{Z}_{n}$ has $2^{k}$
idempotent elements where $k$ is the number of distinct prime factors of $n$.
Specifically, the ring $\mathbb{Z}_{6}$ is not Boolean but it has $4$
idempotents out of $6$ elements, namely, $\\{0,1,3,4\\}$. Finite local rings
with unity contain no idempotents except the trivial elements $0$ and $1$.

###  2.3 Orthogonal codes

Let $k<n$ be positive integers. A rate-$k/n$ linear code $\mathcal{C}$ of
length $n$ over $R$ is an $R$-submodule of $R^{n}$ given by the set
$\mathcal{C}=\\{v\in R^{n}\;|\;v=uG,u\in R^{k}\\}$, where $G$ is a $k\times n$
matrix whose rows span the elements of $\mathcal{C}$ which are called
codewords. If no proper subset of the rows of $G$ generates $\mathcal{C}$,
then $G$ is called a generator matrix of $\mathcal{C}$. The code $\mathcal{C}$
is a leading-systematic code if it has a systematic generator matrix of the
form $G=[I_{k}:A]$, where $I_{k}$ is the $k\times k$ identity matrix and $A$
is a $k\times(n-k)$ matrix, in which case, $\mathcal{C}$ is a free code since
the rows of $G$ form a basis for $\mathcal{C}$. We recall that $\mathcal{C}$
is a weakly self-dual code if $\mathcal{C}\subseteq\mathcal{C}^{\perp}$, where
$\mathcal{C}^{\perp}=\\{u\in R^{n}\;|\;\langle u,v\rangle=0,\;\text{for
all}\;v\in\mathcal{C}\\}$ is the orthogonal or dual code of $\mathcal{C}$, and
that $\mathcal{C}$ is self-dual if $\mathcal{C}=\mathcal{C}^{\perp}$.

Massey in [5] gives specific notions of orthogonal matrices over fields
depending on the result of the product of a given matrix with its transpose.
An $n\times n$ matrix $O$ is an antiorthogonal matrix if $O^{T}O=-I_{n}$, and
is a self-orthogonal matrix if $O^{T}O=0_{n}$, where $0_{n}$ is the all-zero
matrix. Orthogonal, antiorthogonal and self-orthogonal matrices are then used
to construct certain linear codes over fields. It is shown in [5] that a
leading-systematic linear code is self-dual if and only if the matrix $A$ in
its generator matrix $G=[I:A]$ is an antiorthogonal matrix. The conditions for
the existence of antiorthogonal matrices over a Galois field are stated in
[6].

A linear complementary dual code $\mathcal{C}$ (or LCD code) is a code
satisfying $\mathcal{C}\cap\mathcal{C}^{\perp}=\\{\mathbf{0}\\}$. Given the
generator matrix $G$ of a code $\mathcal{C}$, then $\mathcal{C}$ is self-dual
if and only if $GG^{T}$ is a nonsingular matrix, as shown in [7] for the field
case. Moreover, a leading systematic linear code with $G=[I:A]$ is an LCD code
if $A$ is a row self-orthogonal matrix, or equivalently, if $G$ is a row-
orthogonal matrix.

##  3 Results and discussion

We extend the theory of orthogonal matrices by considering left or right sided
$k$-orthogonal matrices over a finite commutative ring $R$ with unity, where
$k$ is some fixed element of $R$. We begin with the structure of one-sided
$k$-orthogonal matrices in $M_{n}(R)$.

###  3.1 $k$-orthogonal matrices

###### Definition 3.1

Let $k$ be a fixed element of $R$. A matrix $A\in M_{n}(R)$ is said to be left
[resp. right] $k$-orthogonal if and only if $A^{T}A=kI_{n}$ [resp.
$AA^{T}=kI_{n}$], where $I_{n}$ is the identity of $M_{n}(R)$. A matrix $A$ is
said to be two-sided $k$-orthogonal (or simply, $k$-orthogonal) if it is both
left $k$-orthogonal and right $k$-orthogonal.

###### Example 3.2

Given the matrix $A=\begin{bmatrix}2&5\\\ 1&2\end{bmatrix}\in
M_{2}(\mathbb{Z}_{6})$. Then $A^{T}A=AA^{T}=\begin{bmatrix}5&0\\\
0&5\end{bmatrix}$ so that $A$ is a two-sided $5$-orthogonal matrix.

Obviously the transpose of a left [resp. right] $k$-orthogonal matrix is right
[resp. left] $k$-orthogonal. Further, if $A$ is left [resp. right]
$k$-orthogonal, then $[\det(A)]^{2}=k^{n}$, and specifically, if $R$ is
Boolean, $\det(A)=k$. However, a left [resp. right] $k$-orthogonal matrix is
not necessarily right [resp. left] $k$-orthogonal, as given in the following
example.

###### Example 3.3

Throughout this paper we will denote by $\mathcal{R}_{2}$ the Boolean ring
$\mathbb{F}_{2}+v\mathbb{F}_{2}=\\{0,1,v,1+v\\}$, where $v^{2}=v$. Given
$A=\begin{bmatrix}v&0\\\ 1+v&1\end{bmatrix}\in M_{2}(\mathcal{R}_{2})$, then
$AA^{T}=\begin{bmatrix}v&0\\\ 0&v\end{bmatrix}$ so that $A$ is right
$v$-orthogonal. The determinant of $A$ is $v$, that is, $A$ is not a unit in
$M_{2}(\mathcal{R}_{2})$. Furthermore, $A^{T}A=\begin{bmatrix}1&1+v\\\
1+v&1\end{bmatrix}$, thus $A$ is not left $v$-orthogonal.

###### Example 3.4

The matrix $A=\begin{bmatrix}1+v&1\\\ 1&1+v\end{bmatrix}\in
M_{2}(\mathcal{R}_{2})$ satisfies $A^{T}A=AA^{T}=vI_{2}$ and hence is two-
sided $v$-orthogonal.

###### Example 3.5

The matrix $B=\begin{bmatrix}1+v&0&1+v\\\ 1&v&1+v\\\ v&v&0\end{bmatrix}\in
M_{3}(\mathcal{R}_{2})$ satisfies $B^{T}B$ $=BB^{T}=0_{3}$, where $0_{3}$ is
the $3\times 3$ all-zero matrix. Therefore, $B$ is a two-sided $0$-orthogonal
matrix.

It is important to note that, if $k\in R$ is idempotent, the scalar matrix
$kI_{n}$ is both left $k$-orthogonal and right $k$-orthogonal. The statement
is false when $k$ is not idempotent. For instance, the matrix $2I_{n}\in
M_{n}(\mathbb{Z}_{6})$. One can easily verify that this matrix is neither left
$2$-orthogonal nor right $2$-orthogonal. We extend this statement in the
proposition below.

###### Proposition 3.6

Let $k$ be an idempotent element. If $A$ is left [resp. right] $k$-orthogonal,
then $kA$ is also left [resp. right] $k$-orthogonal, but not conversely.

Proof: Quite easily, $(kA)^{T}(kA)=k^{2n}A^{T}A=k^{2n+1}I_{n}=kI_{n}.$ $\Box$

###### Example 3.7

The matrix $A=\begin{bmatrix}0&1+v\\\ 1&v\end{bmatrix}$ is right
$(1+v)$-orthogonal. Then $(1+v)A=\begin{bmatrix}0&1+v\\\ 1+v&0\end{bmatrix}$
is also right $(1+v)$-orthogonal. However, the reversal matrix
$\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}$ is not right $(1+v)$-orthogonal.

We then construct the set of left $k$-orthogonal matrices and the set of right
$k$-orthogonal matrices, and denote these sets by $LO_{n}(k,R)$ and
$RO_{n}(k,R)$, respectively. When $k$ is idempotent, these sets are
multiplicative semigroups.

###### Proposition 3.8

Let $k$ be an idempotent element of $R$, and $n$ be a positive integer, $n\geq
2$. The sets $LO_{n}(k,R)$ and $RO_{n}(k,R)$ of left and right $k$-orthogonal
matrices in $M_{n}(R)$, respectively, are semigroups under matrix
multiplication.

Proof: The set $LO_{n}(k,R)=\\{A\in M_{n}(R)\;|\;A^{T}A=kI_{n}\\}$ is nonempty
because the scalar matrix $kI_{n}$ is an element. Since $LO_{n}(k,R)\subseteq
M_{n}(R)$, associativity is inherent. We just need to show that $LO_{n}(k,R)$
is closed under matrix multiplication. Let $A,B\in LO_{n}(k,R).$ Then
$(AB)^{T}(AB)$ $=(B^{T}A^{T})(AB)$ $=B^{T}(kI_{n})B$ $=k(B^{T}B)$
$=k(kI_{n})$$=k^{2}I_{n}$ $=kI_{n}$. The proof for $RO_{n}(k,R)$ is analogous.
$\Box$ The identity $I_{n}$ of $M_{n}(R)$ is not in $LO_{n}(k,R)$ and in
$RO_{n}(k,R)$ in general and hence these semigroups are not monoids. The set
of two-sided $k$-orthogonal matrices, denoted $O_{n}(k,R)$, satisfies by
definition,

$O_{n}(k,R)=LO_{n}(k,R)\cap RO_{n}(k,R)$ (3)

and therefore is a nonempty multiplicative semigroup as well, as $kI_{n}\in
O_{n}(k,R)$. The semigroup $O_{n}(1_{R},R)$ is actually the orthogonal group
$O_{n}(R)$.

###### Example 3.9

The orthogonal group $O_{2}(1,\mathbb{Z}_{6})$ contains exactly 16 two-sided
$1$-orthogonal matrices $A$ with determinant $1$ or $5$, the only units of
$\mathbb{Z}_{6}$, so that $[det(A)]^{2}=1$. $O_{2}(1,\mathbb{Z}_{6})$ is a
(not normal) subgroup of $GL_{2}(\mathbb{Z}_{6})$ with 288 elements.

###### Proposition 3.10

Let $k$ and $k^{\prime}$ be idempotent elements of $R$, and $n\geq 2$ an
integer. Then, either $LO_{n}(k,R)\cap LO_{n}(k^{\prime},R)=\varnothing$ or
$LO_{n}(k,R)=LO_{n}(k^{\prime},R).$ Similarly for the right $k$-orthogonal
semigroups.

Proof: Suppose $LO_{n}(k,R)\cap LO_{n}(k^{\prime},R)\neq\varnothing$. Then
there is a matrix $A$ such that $A\in LO_{n}(k,R)\cap LO_{n}(k^{\prime},R)$.
It follows that $A^{T}A=kI_{n}$ and $A^{T}A=k^{\prime}I_{n}$ and thus
$kI_{n}=k^{\prime}I_{n}$ so that $k=k^{\prime}$. $\Box$

###### Example 3.11

The left orthogonal semigroup $LO_{2}(1+v,\mathcal{R}_{2})$ consists of the
matrices

$\begin{bmatrix}1+v&0\\\ 0&1+v\end{bmatrix},\begin{bmatrix}1&v\\\
v&1\end{bmatrix},\begin{bmatrix}0&1+v\\\
1+v&0\end{bmatrix},\begin{bmatrix}v&1\\\ 1&v\end{bmatrix},$
$\begin{bmatrix}1&0\\\ v&1+v\end{bmatrix},\begin{bmatrix}v&1+v\\\
1&0\end{bmatrix},\begin{bmatrix}0&1\\\
1+v&v\end{bmatrix},\begin{bmatrix}1+v&v\\\ 0&1\end{bmatrix}.$

You will see in Table 1 that $LO_{2}(1+v,\mathcal{R}_{2})\cap
LO_{2}(v,\mathcal{R}_{2})=\varnothing$.

We use the fact that the transpose of a left [resp. right] $k$-orthogonal
matrix is right [resp. left] $k$-orthogonal to claim the following theorem.

###### Theorem 3.12

For any idempotent element $k\in R$ and positive integer $n\geq 2$, we have
$\lvert LO_{n}(k,R)\rvert=\lvert RO_{n}(k,R)\rvert$.

Proof: The map $f:LO_{n}(k,R)\longrightarrow RO_{n}(k,R)$ defined by

$A\mapsto A^{T}$ (4)

is a bijection. $\Box$ This simple correspondence implies that
$LO_{n}(k,R)\neq RO_{n}(k,R)$, even if $R$ is a field. In Section 3.2, we
shall give the formulas for these cardinalities when $R$ is finite. Table 1
illustrates Theorem 3.12 with one-sided orthogonal semigroups
$LO_{2}(v,\mathcal{R}_{2}$) and $RO_{2}(v,\mathcal{R}_{2})$ consisting of
eight matrices each via the correspondence in (4). Since $\mathcal{R}_{2}$ is
Boolean, each left [resp. right] $v$-orthogonal matrix has determinant $v$.
The matrices in the first four rows are the elements of the two-sided
orthogonal semigroup $O_{2}(v,\mathcal{R}_{2})$ which are circulant symmetric.
We prove this observation in a more general sense in Proposition 3.13.

Table 1: Left and right $v$-orthogonal semigroups over $\mathcal{R}_{2}=\mathbb{F}_{2}+v\mathbb{F}_{2}$ $LO_{2}(v,\mathcal{R}_{2})$ | $RO_{2}(v,\mathcal{R}_{2})$
---|---
$\begin{bmatrix}v&0\\\ 0&v\end{bmatrix}$ | $\begin{bmatrix}v&0\\\ 0&v\end{bmatrix}$
$\begin{bmatrix}1&1+v\\\ 1+v&1\end{bmatrix}$ | $\begin{bmatrix}1&1+v\\\ 1+v&1\end{bmatrix}$
$\begin{bmatrix}0&v\\\ v&0\end{bmatrix}$ | $\begin{bmatrix}0&v\\\ v&0\end{bmatrix}$
$\begin{bmatrix}1+v&1\\\ 1&1+v\end{bmatrix}$ | $\begin{bmatrix}1+v&1\\\ 1&1+v\end{bmatrix}$
$\begin{bmatrix}1&0\\\ 1+v&v\end{bmatrix}$ | $\begin{bmatrix}1&1+v\\\ 0&v\end{bmatrix}$
$\begin{bmatrix}v&1+v\\\ 0&1\end{bmatrix}$ | $\begin{bmatrix}v&0\\\ 1+v&1\end{bmatrix}$
$\begin{bmatrix}0&1\\\ v&1+v\end{bmatrix}$ | $\begin{bmatrix}0&v\\\ 1&1+v\end{bmatrix}$
$\begin{bmatrix}1+v&v\\\ 1&0\end{bmatrix}$ | $\begin{bmatrix}1+v&1\\\ v&0\end{bmatrix}$

###### Proposition 3.13

For any $k\in\mathcal{R}_{2}$, $A\in M_{2}(\mathcal{R}_{2})$ is two-sided
$k$-orthogonal if and only if $A$ can be expressed in the form

$A=\left(\begin{matrix}a&b\\\ b&a\end{matrix}\right)$

where $a+b=k$. Consequently, $|O_{2}(k,\mathcal{R}_{2})|=4$.

Proof: The proof is tedious but rather straightforward, by using the fact that
every element of $\mathcal{R}_{2}$ is idempotent. $\Box$

###### Example 3.14

The two-sided $k$-orthogonal semigroups in $M_{2}(\mathcal{R}_{2})$ are

$O_{2}(0,\mathcal{R}_{2})=\left\\{\begin{bmatrix}0&0\\\
0&0\end{bmatrix},\begin{bmatrix}1&1\\\ 1&1\end{bmatrix},\begin{bmatrix}v&v\\\
v&v\end{bmatrix},\begin{bmatrix}1+v&1+v\\\ 1+v&1+v\end{bmatrix}\right\\}$
$O_{2}(1,\mathcal{R}_{2})=\left\\{\begin{bmatrix}1&0\\\
0&1\end{bmatrix},\begin{bmatrix}0&1\\\
1&0\end{bmatrix},\begin{bmatrix}v&1+v\\\
1+v&v\end{bmatrix},\begin{bmatrix}1+v&v\\\ v&1+v\end{bmatrix}\right\\}$
$O_{2}(v,\mathcal{R}_{2})=\left\\{\begin{bmatrix}v&0\\\
0&v\end{bmatrix},\begin{bmatrix}1&1+v\\\
1+v&1\end{bmatrix},\begin{bmatrix}0&v\\\
v&0\end{bmatrix},\begin{bmatrix}1+v&1\\\ 1&1+v\end{bmatrix}\right\\}$
$O_{2}(1+v,\mathcal{R}_{2})=\left\\{\begin{bmatrix}1+v&0\\\
0&1+v\end{bmatrix},\begin{bmatrix}1&v\\\
v&1\end{bmatrix},\begin{bmatrix}0&1+v\\\
1+v&0\end{bmatrix},\begin{bmatrix}1+v&1\\\ 1&1+v\end{bmatrix}\right\\}$

These semigroups can actually be generated via $X+kJ_{2}$, where $X\in
O_{2}(0,\mathcal{R}_{2})$ and $J_{2}$ is the reversal matrix. Note that
$O_{2}(1,\mathcal{R}_{2})$ is a Klein-$4$ group.

In the classical case of a field $F$, a matrix in $M_{n}(F)$ is left
$1$-orthogonal if and only if it is right $1$-orthogonal, as indicated in (1),
that is, $LO_{n}(1,F)=RO_{n}(1,F)=O_{n}(F)$, the group of two-sided orthogonal
$n\times n$ matrices over $F$. However still, $LO_{n}(0,\mathbb{F})\neq
RO_{n}(0,F)$ as given in the example below. We extend this result further from
fields to rings in Theorem 3.16.

###### Example 3.15

Let $\mathbb{F}_{2}$ be the binary field. The one-sided $0$-orthogonal
semigroups in $M_{2}(\mathbb{F}_{2})$ are

$LO_{2}(0,\mathbb{F}_{2})=\left\\{\begin{bmatrix}0&0\\\
0&0\end{bmatrix},\begin{bmatrix}1&1\\\ 1&1\end{bmatrix},\begin{bmatrix}0&1\\\
0&1\end{bmatrix},\begin{bmatrix}1&0\\\ 1&0\end{bmatrix}\right\\}$
$RO_{2}(0,\mathbb{F}_{2})=\left\\{\begin{bmatrix}0&0\\\
0&0\end{bmatrix},\begin{bmatrix}1&1\\\ 1&1\end{bmatrix},\begin{bmatrix}0&0\\\
1&1\end{bmatrix},\begin{bmatrix}1&1\\\ 0&0\end{bmatrix}\right\\}$

It follows that $O_{2}(0,\mathbb{F}_{2})=\left\\{\begin{bmatrix}0&0\\\
0&0\end{bmatrix},\begin{bmatrix}1&1\\\ 1&1\end{bmatrix}\right\\}.$

###### Theorem 3.16

For any commutative ring $R$ with unity $1\neq 0$ and positive integer $n\geq
2$, the multiplicative semigroups $LO_{n}(1,R)$ and $RO_{n}(1_{R},R)$ are
groups.

Proof: Let the set $LO_{n}(1,R)=\\{A\in M_{n}(R)\;|\;A^{T}A=I_{n}\\}$.
Clearly, $I_{n}\in LO_{n}(1,R)$ and is the two-sided identity element, in
particular, $I_{n}$ is the left identity element. Let $A\in LO_{n}(1,R)$.
Since $A^{T}A=I_{n}$, then $A^{T}$ is the left inverse of $A$. Hence
$LO_{n}(1_{R},R)$ contains a left identity element and each element has a left
inverse. Therefore, $LO_{n}(1,R)$ is group. The proof for $RO_{n}(1,R)$ is
analogous. $\Box$

###### Corollary 3.17

For any commutative ring $R$ with unity $1\neq 0$ and positive integer $n\geq
2$, we have $LO_{n}(1,R)=RO_{n}(1,R)=O_{n}(R)$.

Proof: Suppose $A\in LO_{n}(1,R)$. Then $A^{T}A=I_{n}$. Now we take the
product $(AA^{T})(AA^{T})=A(A^{T}A)A^{T}=A(I_{n})A^{T}=AA^{T}$. Since
$LO_{n}(1,R)$ is a group, then by cancellation, $AA^{T}=I_{n}$ so that $A\in
RO_{n}(1,R)$. Hence, $LO_{n}(1,R)\subseteq RO_{n}(1,R)$. Similarly,
$RO_{n}(1,R)\subseteq LO_{n}(1,R)$ . The second equality is a mere consequence
of (3). $\Box$ The converse of Theorem 3.16 is obviously true, as shown below.

###### Proposition 3.18

Let $R$ be any commutative ring with unity $1\neq 0$. If $O_{n}(k,R)$ is a
group, then $k=1$.

Proof: Since $O_{n}(k,R)$ is a subgroup of $GL_{n}(R)$, necessarily the matrix
$I_{n}$, which is in $O_{n}(1,R)$, is also the identity of $O_{n}(k,R)$, and
we are done. $\Box$

###  3.2 Semigroup isomorphisms

Analogous semigroup isomorphisms for the left [resp. right] $k$-orthogonal
semigroups $LO_{n}(k,R)$ [resp. $RO_{n}(k,R)$] are deduced and specific
examples of these isomorphisms are presented in this section. But initially,
we give an easy but useful corollary to Theorem 2.1. On the side, we give a
product formula for the cardinality of $O_{n}(R)$.

###### Corollary 3.19

For any positive integers $m,n\geq 2$, if $R$ is a finite semi-local ring with
maximal ideals $I_{1},I_{2},\ldots,I_{m}$, then

$O_{n}(R)\cong O_{n}(\mathbb{F}_{q_{1}})\times
O_{n}(\mathbb{F}_{q_{2}})\times\ldots\times O_{n}(\mathbb{F}_{q_{m}})$ (5)

where $\mathbb{F}_{q_{j}}$ are Galois fields of order $q_{j}$,
$j=1,2,\ldots,m$.

Proof: In Theorem 2.1, $GL_{n}(R)$ is isomorphic to the product
$GL_{n}(R/I_{1})\times GL_{n}(R/I_{2})\times...\times GL_{n}(R/I_{m})$. Since
$R$ is finite and $I_{j}$ is maximal, then $R/I_{j}$ is a Galois field
$\mathbb{F}_{q_{j}},$ for some $q_{j}$ a power of a prime. That is,

$GL_{n}(R)\cong GL_{n}(\mathbb{F}_{q_{1}})\times
GL_{n}(\mathbb{F}_{q_{2}})\times\ldots\times GL_{n}(\mathbb{F}_{q_{m}})$

Now, $O_{n}(R)$ is a subgroup of $GL_{n}(R)$, and $O_{n}(\mathbb{F}_{q_{j}})$
is a subgroup of $GL_{n}(\mathbb{F}_{q_{j}})$, for $j=1,2,\ldots,m$. By
extension and the correspondence theorem, the result (5) follows. $\Box$
Consequently the number of $n\times n$ orthogonal matrices over $R$ is given
by

$|O_{n}(R)|=|O_{n}(\mathbb{F}_{q_{1}})|\cdot|O_{n}(\mathbb{F}_{q_{2}})|\cdot....\cdot|O_{n}(\mathbb{F}_{q_{m}})|$
(6)

where each factor in the product is computed using the formula from [1].

###### Example 3.20

We have $GL_{2}(\mathcal{R}_{2})\cong GL_{2}(\mathbb{F}_{2})\times
GL_{2}(\mathbb{F}_{2})$ Using the formula in (2) we get
$|GL_{2}(\mathcal{R}_{2})|=6\cdot 6=36$. Further,
$O_{2}(1,\mathcal{R}_{2})\cong O_{2}(\mathbb{F}_{2})\times
O_{2}(\mathbb{F}_{2})$, which is a Klein-$4$ group. Since
$|O_{2}(\mathbb{F}_{2})|=2$ from [1], therefore
$|O_{2}(1,\mathcal{R}_{2})|=4$, as we already know.

###### Example 3.21

We have $O_{2}(1,\mathbb{Z}_{6})\cong O_{2}(\mathbb{F}_{2})\times
O_{2}(\mathbb{F}_{3})$ Now, $|O_{2}(\mathbb{F}_{2})|=2$ and
$|O_{2}(\mathbb{F}_{3})|=8$ from the formulas in [1]. Hence, $\lvert
O_{2}(1,\mathbb{Z}_{6})\rvert=16$ as earlier mentioned in Example 3.9. In
fact, $O_{2}(1,\mathbb{Z}_{6})\cong\mathbb{Z}_{2}\times D_{4}$, where $D_{4}$
is the dihedral group of degree $4$.

###### Corollary 3.22

Let $R$ be a finite semi-local ring and $k\neq 1$ be an idempotent element of
$R$. For positive integers $m,n\geq 2$, we have

$LO_{n}(k,R)\cong LO_{n}(a_{1},\mathbb{F}_{q_{1}})\times
LO_{n}(a_{2},\mathbb{F}_{q_{2}})\times\ldots\times
LO_{n}(a_{m},\mathbb{F}_{q_{m}})$ (7)

where the $q_{i}$’s are powers of primes, and the $a_{i}$’s are determined by
the mapping from $R$ to
$\mathbb{F}_{q_{1}}\times\mathbb{F}_{q_{2}}\times\ldots\times\mathbb{F}_{q_{m}}$.
Analogously for $RO_{n}(k,R)$.

Proof: We consider the isomorphism $\theta$ in Theorem 2.1 as an isomorphism
of multiplicative semigroups and follow the same reasoning in the previous
corollary. Since $R$ is finite and $I_{j}$ is maximal, then $R/I_{j}$ is a
Galois field $\mathbb{F}_{q_{j}},$ for some $q_{j}$ a power of a prime. Now,
$LO_{n}(k,R)$ is a subsemigroup of $M_{n}(R)$, and
$LO_{n}(a_{j},\mathbb{F}_{q_{j}})$ is a subsemigroup of
$M_{n}(\mathbb{F}_{q_{j}})$ for $j=1,\dots,m$. By extension and the
correspondence theorem for semigroups, the result (7) follows. $\Box$
Consequently the number of left $k$-orthogonal matrices over $R$ is given by

$|LO_{n}(R)|=|LO_{n}(a_{1},\mathbb{F}_{q_{1}})|\cdot|LO_{n}(a_{2},\mathbb{F}_{q_{2}})|\cdot....\cdot|LO_{n}(a_{m},\mathbb{F}_{q_{m}})|$
(8)

###### Example 3.23

The Boolean ring $\mathcal{R}_{2}=\mathbb{F}_{2}+v\mathbb{F}_{2}$, $v^{2}=v$
is isomorphic to the product $\mathbb{F}_{2}\times\mathbb{F}_{2}$ via the map
$a+vb\mapsto(a+b,a)$. Hence,

$LO_{n}(a+vb,\mathcal{R}_{2})\cong LO_{n}(a+b,\mathbb{F}_{2})\times
LO_{n}(a,\mathbb{F}_{2}).$

In particular, we have the following isomorphisms.

$LO_{n}(0,\mathbb{F}_{2}+v\mathbb{F}_{2})\cong LO_{n}(0,\mathbb{F}_{2})\times
LO_{n}(0,\mathbb{F}_{2})$ $LO_{n}(v,\mathbb{F}_{2}+v\mathbb{F}_{2})\cong
O_{n}(\mathbb{F}_{2})\times LO_{n}(0,\mathbb{F}_{2})$
$LO_{n}(1+v,\mathbb{F}_{2}+v\mathbb{F}_{2})\cong
LO_{n}(0,\mathbb{F}_{2})\times O_{n}(\mathbb{F}_{2})$

###### Example 3.24

The integer ring $\mathbb{Z}_{6}$ is isomorphic to
$\mathbb{F}_{2}\times\mathbb{F}_{3}$ via the map $x\mapsto(x\mod 2,x\mod 3)$.
It follows that

$LO_{n}(0,\mathbb{Z}_{6})\cong LO_{n}(0,\mathbb{F}_{2})\times
LO_{n}(0,\mathbb{F}_{3})$ $LO_{n}(3,\mathbb{Z}_{6})\cong
O_{n}(\mathbb{F}_{2})\times LO_{n}(0,\mathbb{F}_{3})$
$LO_{n}(4,\mathbb{Z}_{6})\cong LO_{n}(0,\mathbb{F}_{2})\times
O_{n}(\mathbb{F}_{3})$

###### Example 3.25

The set
$\mathbb{F}_{2^{r}}+v\mathbb{F}_{2^{r}}=\\{a+vb\;|\;a,b\in\mathbb{F}_{2^{r}},\;v^{2}=v\\}$
is a commutative semi-local ring with unity under addition and multiplication
modulo $2$. This ring has exactly two proper nontrivial ideals $\langle
v\rangle=\\{xv\;|\;x\in\mathbb{F}_{2^{r}}\\}$ and $\langle
1+v\rangle=\\{x(1+v)\;|\;x\in\mathbb{F}_{2^{r}}\\}$. These ideals are both
maximal and $|\langle v\rangle|=|\langle 1+v\rangle|=2^{r}$. There are
$2(2^{r}-1)$ zero divisors and $(2^{r}-1)^{2}$ units. Now,
$\mathbb{F}_{2^{r}}+v\mathbb{F}_{2^{r}}\cong\mathbb{F}_{2^{r}}\times\mathbb{F}_{2^{r}}$
via the map $a+vb\mapsto(a+b,a)$ so that
$LO_{n}(a+vb,\mathbb{F}_{2^{r}}+v\mathbb{F}_{2^{r}})\cong
LO_{n}(a+b,\mathbb{F}_{2^{r}})\times LO_{n}(a,\mathbb{F}_{2^{r}}).$

###### Example 3.26

For $p$ prime, $p\neq 2$, the set
$\mathbb{F}_{p^{r}}+v\mathbb{F}_{p^{r}}=\\{a+vb\;|\;a,b\in\mathbb{F}_{p^{r}},\;v^{2}=1\\}$
is a commutative semi-local ring with unity under addition and multiplication
modulo $p$. It has exactly two maximal ideals $\langle
1+v\rangle=\\{x(1+v)\;|\;x\in\mathbb{F}_{p^{r}}\\}$ and $\langle
1-v\rangle=\\{x(1-v)\;|\;x\in\mathbb{F}_{p^{r}}\\}$ with $|\langle
1+v\rangle|=|\langle 1-v\rangle|=p^{r}$. There are $2(p^{r}-1)$ zero divisors
and $(p^{r}-1)^{2}$ units. Now,
$\mathbb{F}_{p^{r}}+v\mathbb{F}_{p^{r}}\cong\mathbb{F}_{p^{r}}\times\mathbb{F}_{p^{r}}$
via the map $a+vb\mapsto(a-b,a+b)$ so that
$LO_{n}(a+vb,\mathbb{F}_{p^{r}}+v\mathbb{F}_{p^{r}})\cong
LO_{n}(a+b,\mathbb{F}_{p^{r}})\times LO_{n}(a-b,\mathbb{F}_{p^{r}}).$

We summarize in the tables that follow the cardinalities of the finite
semigroups of left and right $k$-orthogonal matrices over $\mathcal{R}_{2}$
and $\mathbb{Z}_{6}$. MAGMA® programs are written to create the semi-local
ring and identify its idempotents, then for $n=2,3$ and for each idempotent
$k$, elements of the matrix ring are exhaustively tested and counted for the
semigroup. The resulting cardinalities are compared with the product formulas
derived above. With slight adjustments, the routines can work for other rings
as well.

Table 2: $k$-Orthogonal $2\times 2$ Matrices Over $\mathcal{R}_{2}$ $k$ | $LO_{2}(k,\mathcal{R}_{2})=RO_{2}(k,\mathcal{R}_{2})$ | $O_{2}(k,\mathcal{R}_{2})$ | $LO_{2}(k,\mathcal{R}_{2})-O_{2}(k,\mathcal{R}_{2})$
---|---|---|---
$0$ | $16$ | $4$ | $12$
$1$ | $4$ | $4$ | $0$
$v$ | $8$ | $4$ | $4$
$1+v$ | $8$ | $4$ | $4$
Table 3: $k$-Orthogonal $3\times 3$ Matrices Over $\mathcal{R}_{2}$ $k$ | $LO_{3}(k,\mathcal{R}_{2})=RO_{3}(k,\mathcal{R}_{2})$ | $O_{3}(k,\mathcal{R}_{2})$ | $LO_{3}(k,\mathcal{R}_{2})-O_{3}(k,\mathcal{R}_{2})$
---|---|---|---
$0$ | $484$ | $100$ | $384$
$1$ | $36$ | $36$ | $0$
$v$ | $132$ | $60$ | $72$
$1+v$ | $132$ | $60$ | $72$
Table 4: $k$-Orthogonal $2\times 2$ Matrices Over $\mathbb{Z}_{6}$ $\;\;\;k\;\;\;$ | $LO_{2}(k,\mathbb{Z}_{6})=RO_{2}(k,\mathbb{Z}_{6})$ | $O_{2}(k,\mathbb{Z}_{6})$ | $LO_{2}(k,\mathbb{Z}_{6})-O_{2}(k,\mathbb{Z}_{6})$
---|---|---|---
$0$ | $4$ | $2$ | $2$
$1$ | $16$ | $16$ | $0$
$3$ | $2$ | $2$ | $0$
$4$ | $32$ | $16$ | $16$
Table 5: $k$-Orthogonal $3\times 3$ Matrices Over $\mathbb{Z}_{6}$ $\;\;\;k\;\;\;$ | $LO_{3}(k,\mathbb{Z}_{6})=RO_{3}(k,\mathbb{Z}_{6})$ | $O_{3}(k,\mathbb{Z}_{6})$ | $LO_{3}(k,\mathbb{Z}_{6})-O_{3}(k,\mathbb{Z}_{6})$
---|---|---|---
$0$ | $2310$ | $330$ | $1980$
$1$ | $288$ | $288$ | $0$
$3$ | $630$ | $198$ | $432$
$4$ | $1056$ | $480$ | $576$

###  3.3 Codes from $k$-orthogonal matrices

We study how $(-1)$-orthogonal matrices and $0$-orthogonal matrices can give
rise to leading-systematic self-dual or weakly self-dual linear codes. The
construction of these classes of codes is illustrated in some examples.

In this section, we let $R$ to be a finite commutative ring with unity.
Following the discussion in the previous section, we use the terminology in
[5] to say that a matrix $A\in M_{n}(R)$ is left [resp. right] antiorthogonal
if and only if $A$ is left [resp. right] $(-1)$-orthogonal. If $A$ is both
left antiorthogonal and right antiorthogonal, then $A$ is said to be an
antiorthogonal matrix. However, the following proposition says that one-sided
antiorthogonal matrices over $R$ coincide.

###### Proposition 3.27

A left antiorthogonal matrix is a right antiorthogonal matrix, and conversely.

Proof: Suppose $AA^{T}=-I_{n}$. Then $\det A$ is a unit in $R$ so that $A\in
GL_{n}(R)$ in which left and right inverses coincide. The result
follows.$\Box$ The matrix in Example 3.2 is an antiorthogonal matrix. Clearly,
an antiorthogonal matrix is orthogonal if and only if the characteristic of
$R$ is $2$. Consequently, Proposition $1$ of [5] can be stated more generally
as follows.

###### Proposition 3.28

Let $R$ be a finite commutative ring with unity. A rate-$k/n$ leading-
systematic linear code over $R$ is self-dual if and only if, in its generator
matrix $G=[I_{k}:A]$, the matrix $A$ is antiorthogonal.

It should be noted that, in this case, $n=2k$ so that $A$ is also a $k\times
k$ matrix. The MAGMA® routines that we have developed help us to find the one-
sided $k$-orthogonal matrices, and in particular, the antiorthogonal matrices
which subsequently give the self-dual codes, as illustrated in the example
below. The self-duality is checked further in MAGMA®.

###### Example 3.29

Consider the two-sided $5$-orthogonal matrix $A=\begin{bmatrix}4&5\\\
1&4\end{bmatrix}\in M_{2}(\mathbb{Z}_{6})$. Then form the $2\times 4$
generator matrix

$G=\begin{bmatrix}I_{2}&A\end{bmatrix}=\begin{bmatrix}1&0&4&5\\\
0&1&1&4\end{bmatrix}.$

The leading-systematic code over $\mathbb{Z}_{6}$ generated by $G$ is self-
dual.

###### Example 3.30

The rate-$4/8$ octacode $\mathcal{O}_{8}$ over $\mathbb{Z}_{4}$ is a self-dual
leading-systematic linear code with generator matrix $G=[I_{4}:A]$ where

$A=\begin{bmatrix}3&1&2&1\\\ 1&2&3&1\\\ 3&3&3&2\\\ 2&3&1&1\end{bmatrix}$

is a two-sided $3$-orthogonal matrix. The Lee distance is $6$ and the Hamming
distance is $4$.

The antiorthogonality condition can be extended to non-square matrices. A
$k\times m$ matrix $A$ is said to be right row-antiorthogonal if
$AA^{T}=-I_{k}$. Deleting rows of an antiorthogonal matrix will result to a
right row-antiorthogonal matrix, but not every right row antiorthogonal matrix
can be constructed. We claim an extension of Proposition $2$ in [5] as
follows.

###### Proposition 3.31

Let $R$ be a finite commutative ring with unity. A leading-systematic linear
code over $R$ is weakly self-dual if and only if, in its generator matrix
$G=\begin{bmatrix}I_{k}&A\end{bmatrix}$, the matrix $A$ is right row-
antiorthogonal.

###### Example 3.32

With the matrix $A$ of the octacode $\mathcal{O}_{8}$ in Example 3.30, delete
the fourth row to get $B=\begin{bmatrix}3&1&2&1\\\ 1&2&3&1\\\
3&3&3&2\end{bmatrix}$. It can be verified that $B$ is a right row-
antiorthogonal matrix. We then construct the leading-systematic rate-$3/7$
$\mathbb{Z}_{4}$-linear code using the generator matrix

$G=\begin{bmatrix}I_{3}&B\end{bmatrix}=\begin{bmatrix}1&0&0&3&1&2&1\\\
0&1&0&1&2&3&1\\\ 0&0&1&3&3&3&2\end{bmatrix}.$

The code generated by $G$ is weakly self-dual. The Lee distance is $6$ and the
Hamming distance is $4$.

A matrix $A\in M_{n}(R)$ is said to be left [resp. right] self-orthogonal if
and only if $A$ is left [resp. right] $0$-orthogonal. If $A$ is both left
self-orthogonal and right self-orthogonal, then the matrix $A$ is simply
called self-orthogonal. The matrix in Example 3.5 is a self-orthogonal matrix.
Unlike the case of antiorthogonal matrices, left self-orthogonal matrices and
right self-orthogonal matrices do not necessarily coincide.

###### Example 3.33

The matrix in $M_{3}(\mathbb{Z}_{6})$ given by

$A=\begin{bmatrix}0&3&3\\\ 4&2&4\\\ 2&1&5\end{bmatrix}$

is right self-orthogonal but not left self-orthogonal. We adopt the analogous
idea of a row right self-orthogonal matrix. Deleting any row of $A$, say, the
third row, we get the $2\times 3$ matrix $G=\begin{bmatrix}0&3&3\\\
4&2&4\end{bmatrix}$, which is a row right self-orthogonal matrix. The matrix
$G$ generates a weakly self-dual code over $\mathbb{Z}_{6}$.

This example is generalized in the proposition below.

###### Proposition 3.34

Let $R$ be a finite commutative ring with unity. A linear code over $R$ with
generator matrix $G$ is weakly self-dual if and only if $G$ is row right self-
orthogonal.

##  4 Conclusion and recommendations

For further research it is interesting to derive explicit formulas for the
cardinality of the finite one-sided $k$-orthogonal semigroups. It is also
worthy to consider applying the results of this paper to other classes of
finite semi-local rings with multiple idempotents such as those given in
Examples 3.25 and 3.26. More specific and relevant structural properties of
the matrix semigroups can be explored.

We have also proved that there exist no $3\times 3$ antiorthogonal matrices
over the integer ring $\mathbb{Z}_{6}$. This is accomplished by using the map
from $\mathbb{Z}_{6}$ onto $\mathbb{F}_{2}\times\mathbb{F}_{3}$ in Example
3.24 to show that indeed $-1$ is a non-square in $\mathbb{Z}_{6}$, and by [6]
that, for any odd integer $n$, there exist no $n\times n$ antiorthogonal
matrices over $\mathbb{F}_{3}$. This specific result is likewise verified
through exhaustive search. The reader is encouraged to determine the
conditions for the existence of antiorthogonal matrices over finite semi-local
rings.

Finally, the connections of the above defined matrices to the characterization
of LCD codes over finite commutative rings with unity can be studied.

##  5 Acknowledgement

The first author gratefully acknowledges the University of the Philippines for
the One U.P. Professorial Chair Award in Mathematics (Coding Theory) for
Outstanding Research and Teaching in U.P. Los Baños from 2016 to 2021.

## References

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