# DNA Codes over the Ring $\mathbb{Z}_{4}+w\mathbb{Z}_{4}$

Adel Alahmadi11111 Department of Mathematics, King Abdulaziz University,
Jeddah, Saudi Arabia, Krishna Gopal Benerjee22222 Department of Electrical
Engineering, Indian Institute of Technology Kanpur, Uttar Pradesh - 208016,
India, Sourav Deb33333 Dhirubhai Ambani Institute of Information and
Communication Technology Gandhinagar, Gujarat - 382007, India, and
Manish K Gupta3 email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

In this present work, we generalize the study of construction of DNA codes
over the rings $\mathcal{R}_{\theta}=\mathbb{Z}_{4}+w\mathbb{Z}_{4}$,
$w^{2}=\theta$ for $\theta\in\mathbb{Z}_{4}+w\mathbb{Z}_{4}$. Rigorous study
along with characterization of the ring structures is presented. We extend the
Gau map and Gau distance, defined in [1], over all the $16$ rings
$\mathcal{R}_{\theta}$. Furthermore, an isometry between the codes over the
rings $\mathcal{R}_{\theta}$ and the analogous DNA codes is established in
general. Brief study of dual and self dual codes over the rings is given
including the construction of special class of self dual codes that satisfy
reverse and reverse-complement constraints. The technical contributions of
this paper are twofold. Considering the Generalized Gau distance, Sphere
Packing-like bound, GV-like bound, Singleton like bound and Plotkin-like bound
are established over the rings $\mathcal{R}_{\theta}$. In addition to this,
optimal class of codes are provided with respect to Singleton-like bound and
Plotkin-like bound. Moreover, the construction of family of DNA codes is
proposed that satisfies reverse and reverse-complement constraints using the
Reed-Muller type codes over the rings $\mathcal{R}_{\theta}$.

###### Index Terms:

DNA Codes, Ring $\mathbb{Z}_{4}+w\mathbb{Z}_{4}$, DNA Data Storage

##  I Introduction

The topic of DNA codes has fascinated researchers for the last two decades.
The intrinsic idea of computation using DNA has been posed in a seminal paper
by Adleman in 1994 [2]. Driven by the ground-breaking idea, the branch of
mathematical construction for artificial DNA codes and many more
interdisciplinary topics, such as DNA-based data storage systems, computation
using DNA tiles, etc. have emerged. DNA (DeoxyriboNucleic Acid) stores of all
genetic as well as biological instructions of life. DNA strands can be
recognised by four nucleotides (nt) or bases, Adenine ($A$), Guanine ($G$),
Cytosine ($C$) and Thymine ($T$), where the Watson-Crick complements for $A$
and $G$ are $T$ and $C$ respectively, $i.e.$, $A^{c}=T$ and $G^{c}=C$. A DNA
strand is a string of certain length containing the symbols $A$, $G$, $C$ and
$T$, $e.g.$, $ACTTAGA$ is a DNA string of length $7$, as the string contains
$7$ symbols, not necessarily distinct. For a DNA string ${\bf x}=(x_{1}\
x_{2}\ \ldots\ x_{n})$ of length $n$, the reverse and complement DNA strings
are considered as ${\bf x}^{r}=(x_{n}\ x_{n-1}\ \ldots\ x_{1})$ and ${\bf
x}^{c}=(x^{c}_{1}\ x^{c}_{2}\ \ldots\ x^{c}_{n})$, respectively. Following
these definitions, the reverse-complement DNA string of the DNA string ${\bf
x}=(x_{1}\ x_{2}\ \ldots\ x_{n})$ of length $n$ will be ${\bf
x}^{rc}=(x^{c}_{n}\ x^{c}_{n-1}\ \ldots\ x^{c}_{1})$. Considering the
previously stated DNA string ${\bf x}=ACTTAGA$, the corresponding reverse,
complement and reverse-complement DNA strings will be ${\bf x}^{r}=AGATTCA$,
${\bf x}^{c}=TGAATCT$ and ${\bf x}^{rc}=TCTAAGT$, respectively. The key point
of using DNA for computing is the DNA hybridization in between the string(s),
which however is also the source of errors. The purpose of constructing DNA
codes using different approaches is to generate sufficiently dissimilar DNA
strands to avoid error(s) in any aspect [3, 4]. Meanwhile, various celebrated
constructions have been proposed [5], which produce optimal DNA codes
satisfying different constraints, where DNA code construction using rings and
fields is treated as a prominent one. Recently, DNA codes satisfying multiple
constraints are constructed in [6, 7].

We denote the DNA symbol set $\Sigma_{DNA}=\\{A,G,C,T\\}$. Also Hamming
distance $d_{H}$ between two DNA strings ${\bf x}=(x_{1}\ x_{2}\ \ldots\
x_{n})$ and ${\bf y}=(y_{1}\ y_{2}\ \ldots\ y_{n})$ of length $n$ is denoted
as $d_{H}({\bf x},{\bf y})$ to be the number of positions where the strings
${\bf x}$ and ${\bf y}$ differ, $i.e.$, the number of elements in the set
$\\{i:x_{i}\neq y_{i}\\}$, where $i\in\\{1,2,\ldots n\\}$. A DNA code over the
symbol set $\Sigma_{DNA}$ is defined by
$\mathscr{C}_{DNA}(n,M,d_{H})\subseteq\Sigma^{n}_{DNA}$, containing $M$
distinct DNA strings or codewords, each of length $n$ and having minimum
Hamming distance $d_{H}$ between any two codewords. According to the
definition $d_{H}=\min\\{d_{H}({\bf x},{\bf y}):\mbox{for all }{\bf x},{\bf
y}\in\mathscr{C}_{DNA},s.t.,{\bf x}\neq{\bf y}\\}$. In this paper, the DNA
code $\mathscr{C}_{DNA}$ is considered as reversible DNA code or closed under
reverse constraint, if for each ${\bf x}\in\mathscr{C}_{DNA}$, ${\bf
x}^{r}\in\mathscr{C}_{DNA}$. Similarly, a DNA code $\mathscr{C}_{DNA}$ is a
complement DNA code if, for each ${\bf x}\in\mathscr{C}_{DNA}$, ${\bf
x}^{c}\in\mathscr{C}_{DNA}$. We also consider that if, for each ${\bf
x}\in\mathscr{C}_{DNA}$, ${\bf x}^{rc}\in\mathscr{C}_{DNA}$ then the DNA code
$\mathscr{C}_{DNA}$ satisfies the reverse-complement constraint, and the code
is called reversible-complement DNA code. To maximize the output reliability
of DNA codes, a set of combinatorial constraints, $i.e.$, Hamming constraint,
reverse constraint, reverse-complement constraint and $GC$-content constraint,
are taken into consideration in efficient constructions [8]. Thus motivated,
researchers are attracted towards the algebraic construction of DNA codes
where each codeword satisfies the given constraints to ensure the appropriate
match during the process of DNA hybridization and avoid error(s).

Rykov et. al [9] pioneered the construction of DNA codes using sets of
reverse-complement cyclic codes over quaternary alphabets designating $0,1,2$
and $3$ for $A,C,G$ and $T$, respectively. Theoretical bounds are also
established for parameters of the constructed codes in this paper. From the
theoretical point of view, it is quite evident to construct DNA codes over the
ring $\mathbb{Z}_{4}$ and the field $GF(4)$ or its variants. The construction
of linear DNA codes using the ring $\mathbb{Z}_{4}$ was proposed by King et.
al in [3], where the constructed codes follow the reverse and reverse-
complement constraints. A parallel approach can be found in [10] to construct
linear and additive cyclic codes of odd length that satisfy reverse-complement
constraints over the Galois filed $GF(4)$ of the purpose of DNA computing. In
[11, 12], two different approaches can be observed to construct DNA sequences
using BCH codes over $GF(4)$ and $\mathbb{Z}_{4}$-linear codes, respectively.

In this paper, we restrict our focuses to the ring
$\mathbb{Z}_{4}+w\mathbb{Z}_{4}$ and its variants, containing $16$ elements.
The class of finite rings $S_{2m}=\mathbb{Z}_{2m}+i\mathbb{Z}_{2m}$,
$i^{2}=-1$ was introduced in 2005 by Choie and Dougherty in [13] and self dual
codes are studies. Motivated by the approach, in 2014 linear codes as well as
self dual codes which are analogous allowing some projections from the ring
$\mathbb{Z}_{4}+u\mathbb{Z}_{4}$ to $\mathbb{Z}_{4}$ and
$\mathbb{F}_{2}+u\mathbb{F}_{2}$, have been constructed by Yildiz et. al [14].
Furthermore, MacWilliams identities were also established in this paper.
Cyclic DNA codes of odd length over the rings
$\mathbb{Z}_{4}+w\mathbb{Z}_{4}$, $w^{2}=2$ and
$\mathbb{Z}_{4}+u\mathbb{Z}_{4}$, $u^{2}=0$ have been constructed in [15],
[16]. Usage of the ring $\mathbb{F}_{4}+u\mathbb{F}_{4}$, $u^{2}=0$ in
constructing reversible cyclic DNA codes can be found in [17]. In [18], Siap
et. al presented an extensive study on characterizing the structure of linear,
constacyclic and cyclic codes over the ring $\mathbb{F}_{4}[v]/(v^{2}-v)$ and
construction of optimal DNA codes satisfying reverse-complement constraint via
reversible codes over the considered ring was proposed. Ding et. al in [19],
have constructed cyclic DNA codes over the ring $\mathbb{Z}_{4}[u]/(u^{2}-1)$
of odd length based on deletion distance. In [20], self dual codes are
constructed over the rings $\mathbb{Z}_{4}+w\mathbb{Z}_{4}$, $w^{2}=i$ where
$i\in\\{1,2w\\}$. Circulant self-dual codes and Type II codes over these rings
were also studied in this paper. Recently in 2018, Limbachiya $et.\ al$ [1]
defined the Gau distance on the elements of the ring
$\mathbb{Z}_{4}+w\mathbb{Z}_{4}$, $w^{2}=2+2w$ and constructed DNA codes of
even length, which satisfy the reverse and reverse-complement constraints. The
authors also proposed the Gau map, which is an isometry between the ring
elements and their corresponding DNA string of length two. That is, the map
preserves the distance from the codes over the ring based on Gau distance and
the corresponding DNA codes over $\Sigma^{2n}_{DNA}$ based on Hamming
distance.

###  I-A Contributions:

Driven by the construction proposed in [1], we extend the study of codes over
the ring $\mathcal{R}_{\theta}=\mathbb{Z}_{4}+w\mathbb{Z}_{4}$, $w^{2}=\theta$
such that $\theta\in\mathbb{Z}_{4}+w\mathbb{Z}_{4}$ with respect to the
Generalized Gau distance and establish a complete characterization of DNA
codes using Generalized Gau map, where the Generalized Gau distance and
Generalized Gau map are Gau distance and Gau map defined over the ring
$\mathbb{Z}_{4}+w\mathbb{Z}_{4}$ for any
$\theta\in\mathbb{Z}_{4}+w\mathbb{Z}_{4}$. The ring structure classification
provides a symmetric distribution of $8$ chain and $8$ non-chain rings. The
generalized construction proposes the necessary and sufficient condition so
that the DNA codes satisfy the reverse and reverse-complement constraints.
Using the construction, sufficient conditions on different categories of self
dual codes that satisfy the reverse and reverse-complement constraints are
proposed. The present work proposes theoretical bounds on parameters for the
construction. Furthermore, we obtain the parameters of different classes of
optimal codes that satisfy Singleton-like bound and Plotkin-like bound. A
family of DNA codes satisfying reverse and reverse-complement constraints has
been proposed using Reed-Muller type of codes over the ring.

###  I-B Organization:

This paper is organized in the following manner Section II provides a complete
overview of the ring structures $\mathcal{R}_{\theta}$,
$\theta\in\mathbb{Z}_{4}+w\mathbb{Z}_{4}$. The generalization of Gau map and
Gau distance over the ring $\mathcal{R}_{\theta}$ is proposed in Section III.
The necessary and sufficient conditions for reverse and complement codes are
derived and exploited to construct DNA codes using Reed-Muller type of codes.
In Section IV, we discuss the dual and self dual codes over the ring
$\mathcal{R}_{\theta}$ along with the necessary conditions such that a special
class of constructed codes satisfies reverse and complement constraints, which
can be considered as a rich source of efficient DNA codes. Section V includes
theoretical bounds that are established considering the generalized Gau
distance over the rings $\mathcal{R}_{\theta}$ along with the parameters of
families of optimal codes corresponding to certain bounds. Section VI
concludes the paper.

##  II Preliminaries on the rings $\mathcal{R}_{\theta},\ \text{where}\
\theta\in\mathbb{Z}_{4}+w\mathbb{Z}_{4}$

Throughout the paper, we denote
$\mathcal{R}_{\theta}=\mathbb{Z}_{4}+w\mathbb{Z}_{4}$ where
$w^{2}=\theta\in\mathbb{Z}_{4}+w\mathbb{Z}_{4}$. There are $16$ such rings.
These $16$ rings can be classified into two types, $i.e.$, chain rings and
non-chain rings due to their different ring structures. One can observe that,
the division into two classes is symmetric, $i.e.$, $8$ of these rings are
chain rings and the rest are non-chain rings. An explicit classification is
given in Table I for the rings
$\mathcal{R}_{\theta},\theta\in\mathbb{Z}_{4}+w\mathbb{Z}_{4}$.

TABLE I: The classification for the rings $\mathcal{R}_{\theta},w^{2}=\theta$ Chain Rings | $\theta\in\\{2,3,1+w,3+w,1+2w,2+2w,1+3w,3+3w\\}$
---|---
Non-Chain Rings | $\theta\in\\{0,1,w,2w,3w,2+w,3+2w,2+3w\\}$

We discuss the structure of the $8$ finite commutative principal local chain
rings $\mathcal{R}_{\theta},\theta\in\mathbb{Z}_{4}+w\mathbb{Z}_{4}$ in this
section. For given $\theta$, the set of zero divisors and the set of units of
the ring $\mathcal{R}_{\theta}$ are denoted as $\mathcal{Z}_{\theta}$ and
$\mathcal{U}_{\theta}$, respectively. The set $\mathcal{R}^{n}_{\theta}$,
along with the operations additions “$+$” and multiplication “.” can be viewed
as a module over the ring $\mathcal{R}_{\theta}$. Some further classifications
can be done in the class of chain rings in terms of basic ring structures.

###  II-A Ring $\mathcal{R}_{\theta},\theta\in\\{1+w,3+w,1+3w,3+3w\\}$.

For $\mathcal{R}_{\theta},\theta\in\\{a+wb:a,b\in
2\mathbb{Z}_{4}+1\\}=\\{1+w,3+w,1+3w,3+3w\\}$, we have,
$\mathcal{Z}_{1+w}=\mathcal{Z}_{3+w}=\mathcal{Z}_{1+3w}=\mathcal{Z}_{3+3w}=\\{a+wb:a,b\in
2\mathbb{Z}_{4}\\}=\\{0,2,2w,2+2w\\}$; and
$\mathcal{U}_{1+w}=\mathcal{U}_{3+w}=\mathcal{U}_{1+3w}=\mathcal{U}_{3+3w}=\\{a+wb:a,b\in\mathbb{Z}_{4},a+b\in
2\mathbb{Z}_{4}+1\\}\cup\\{a+wb:a,b\in 2\mathbb{Z}_{4}+1,a+b\in
2\mathbb{Z}_{4}\\}=\\{1,3,w,3w,1+w,2+w,3+w,1+2w,3+2w,1+3w,2+3w,3+3w\\}$.

The rings satisfy the chain condition with $3$ distinct ideals as, $\langle
0\rangle\subset\langle 2\rangle=\langle 2w\rangle=\langle
2+2w\rangle\subset\mathcal{R}_{\theta}$.

###  II-B Ring $\mathcal{R}_{\theta},\theta\in\\{2,2+2w\\}$.

For $\mathcal{R}_{\theta},\theta\in\\{a+wb:a=2,b\in
2\mathbb{Z}_{4}\\}=\\{2,2+2w\\}$, we have,
$\mathcal{Z}_{2}=\mathcal{Z}_{2+2w}=\\{a+wb:a\in
2\mathbb{Z}_{4},b\in\mathbb{Z}_{4}\\}=\\{0,2,w,2w,3w,2+w,2+2w,2+3w\\}$; and
$\mathcal{U}_{2}=\mathcal{U}_{2+2w}=\\{a+wb:a\in
2\mathbb{Z}_{4}+1,b\in\mathbb{Z}_{4}\\}=\\{1,3,1+w,3+w,1+2w,3+2w,1+3w,3+3w\\}$.

These rings have $5$ distinct ideals as, $\langle 0\rangle\subset\langle
2w\rangle\subset\langle 2\rangle=\langle 2+2w\rangle\subset\langle
w\rangle=\langle 3w\rangle=\langle 2+w\rangle=\langle
2+3w\rangle\subset\mathcal{R}_{\theta}$.

###  II-C Ring $\mathcal{R}_{\theta},\theta\in\\{3,1+2w\\}$.

For $\mathcal{R}_{\theta},\theta\in\\{a+wb:a\in 2\mathbb{Z}_{4}+1,b\in
2\mathbb{Z}_{4},a\neq b+1\\}=\\{3,1+2w\\}$, We have,
$\mathcal{Z}_{3}=\mathcal{Z}_{1+2w}=\\{a+wb:a,b\in\mathbb{Z}_{4},a-b\in
2\mathbb{Z}_{4}\\}=\\{0,2,2w,1+w,3+w,2+2w,1+3w,3+3w\\}$; and
$\mathcal{U}_{3}=\mathcal{U}_{1+2w}=\\{a+wb:a,b\in\mathbb{Z}_{4},a-b\in
2\mathbb{Z}_{4}+1\\}=\\{1,3,w,3w,2+w,1+2w,3+2w,2+3w\\}$.

The rings have $5$ distinct ideals as, $\langle 0\rangle\subset\langle
2+2w\rangle\subset\langle 2\rangle=\langle 2w\rangle\subset\langle
1+w\rangle=\langle 3+w\rangle=\langle 1+3w\rangle=\langle
3+3w\rangle\subset\mathcal{R}_{\theta}$.

Any subset of $\mathcal{R}^{n}_{\theta}$ is defined as a code
$\mathscr{C}_{\theta}(n,M,d)$ over the ring $\mathcal{R}_{\theta}$ (for given
$\theta$), which consists of $M$ distinct codewords, each of length $n$ and
any two distinct codewords differ by distance at least $d$. Any submodule of
the module $\mathcal{R}^{n}_{\theta}$ is considered as a linear code
$\mathscr{C}_{\theta}$ over the ring $\mathcal{R}_{\theta}$. We denote the
generating matrix of the linear code $\mathscr{C}_{\theta}$ over the ring
$\mathcal{R}_{\theta}$ by $G_{\theta}$ and the row span of $G_{\theta}$ is
denoted as $\langle G_{\theta}\rangle$. The generator matrix of the linear
code $\mathscr{C}_{\theta}$ over the ring $\mathcal{R}_{\theta}$ in standard
form, is given by:

  * a)

For $\theta\in\\{1+w,3+w,1+3w,3+3w\\}$, the generating matrix $G_{\theta}$ in
standard form, of the linear code $\mathscr{C}_{\theta}$ over the ring
$\mathcal{R}_{\theta}$ will be,

$G_{\theta}=\left(\begin{array}[]{ccc}I_{k_{0}}&A_{0,1}&A_{0,2}\\\
0&2I_{k_{1}}&2A_{1,2}\\\ \end{array}\right),$ (1)

where the matrices $A_{i,j}$ are defined over the ring $\mathcal{R}_{\theta}$
for $0\leq i<j\leq 2$. The matrix $G_{\theta}$ is called type
$\\{k_{0},k_{1}\\}$, and the linear code with the generator matrix
$G_{\theta}$ is called type $\\{k_{0},k_{1}\\}$ linear code. The total number
of codewords of the type $\\{k_{0},k_{1}\\}$ linear code
$\mathscr{C}_{\theta}$ will be $16^{k_{0}}4^{k_{1}}$, where
$\theta\in\\{1+w,3+w,1+3w,3+3w\\}$.

  * b)

For $\theta\in\\{2,2+2w\\}$, the generating matrix $G_{\theta}$ in standard
form of the linear code $\mathscr{C}_{\theta}$ over the ring
$\mathcal{R}_{\theta}$ will be,

$G_{\theta}=\left(\begin{array}[]{ccccc}I_{k_{0}}&A_{0,1}&A_{0,2}&A_{0,3}&A_{0,4}\\\
0&wI_{k_{1}}&wA_{1,2}&wA_{1,3}&wA_{1,4}\\\ 0&0&2I_{k_{2}}&2A_{2,3}&2A_{2,4}\\\
0&0&0&2wI_{k_{3}}&2wA_{3,4}\\\ \end{array}\right),$ (2)

where the matrices $A_{i,j}$’s are defined over the ring
$\mathcal{R}_{\theta}$ as,
$A_{i,j}=B^{1}_{i,j}+wB^{2}_{i,j}+2B^{3}_{i,j}+2wB^{4}_{i,j}$, where $\
B^{k}_{i,j}$ are binary matrices for $0\leq i<j\leq 4$ and $k=1,2,3,4$. The
linear code $\mathscr{C}_{\theta}$ with the generating matrix in the given
form, is of type $\\{k_{0},k_{1},k_{2},k_{3}\\}$ and has
$16^{k_{0}}8^{k_{1}}4^{k_{2}}2^{k_{3}}$ codewords in total, where
$\theta\in\\{2,2+2w\\}$.

  * c)

For $\theta\in\\{3,1+2w\\}$, the generating matrix $G_{\theta}$ of the linear
code $\mathscr{C}_{\theta}$ over the ring $\mathcal{R}_{\theta}$, in standard
form will be the matrix given in (3), where the matrices $A_{i,j}$’s are
defined over the ring $\mathcal{R}_{\theta}$ as,
$A_{i,j}=B^{1}_{i,j}+(1+w)B^{2}_{i,j}+2B^{3}_{i,j}+(2+2w)B^{4}_{i,j}$, where
$\ B^{k}_{i,j}$ are binary matrices for $0\leq i<j\leq 4$ and $k=1,2,3,4$.
From the generating matrix $G_{\theta}$ in the standard form, it can deduced
that the linear code $\mathscr{C}_{\theta}$ is of type
$\\{k_{0},k_{1},k_{2},k_{3}\\}$ and has
$16^{k_{0}}8^{k_{1}}4^{k_{2}}2^{k_{3}}$ codewords, where
$\theta\in\\{3,1+2w\\}$.

$G_{\theta}=\left(\begin{array}[]{ccccc}I_{k_{0}}&A_{0,1}&A_{0,2}&A_{0,3}&A_{0,4}\\\
0&(1+w)I_{k_{1}}&(1+w)A_{1,2}&(1+w)A_{1,3}&(1+w)A_{1,4}\\\
0&0&2I_{k_{2}}&2A_{2,3}&2A_{2,4}\\\ 0&0&0&(2+2w)I_{k_{3}}&(2+2w)A_{3,4}\\\
\end{array}\right)$ (3)

##  III Generalized Gau Map and Generalized Gau Distance

We define a correspondence between the elements of the ring
$\mathcal{R}_{\theta}$ and the DNA alphabets for the purpose of constructing
DNA codes satisfying different constraints. The correspondence leads to an
isometry (Distance preserving map) between the codes over the ring
$\mathcal{R}_{\theta}$ and the DNA codes in parallel. In the case of DNA
codes, the Hamming Distance has been defined, while for the codes over the
ring $\mathcal{R}_{\theta}$, Generalized Gau Distance $d_{G(\theta)}$ over
$\mathcal{R}_{\theta}$ (eventually over $\mathcal{R}^{n}_{\theta}$) has been
introduced. The isometry has been established simultaneously by defining
Generalized Gau Map $\phi$ over the ring $\mathcal{R}_{\theta}$. One can
verify that $\phi$ is a bijective mapping between the elements of the ring
$\mathcal{R}_{\theta}$ and the pair DNA nucleotides and it will be independent
of the choice of $\theta$. The arrangement of matrix given in (4) is managed
in such a way that the Hamming distance between any two distinct pair of DNA
neucleotides in different rows or different columns is $2$, otherwise it is
$1$ and we fill the elements of the matrix using the elements of the ring
$\mathcal{R}_{\theta}$ according to the conditions $\phi^{-1}(\phi(x)^{r})=3x$
and $\phi^{-1}(\phi(x)^{c})=x+\lambda$, where $\lambda\in\\{2,2w,2+2w\\}$.
Using the conditions we obtain the following restrictions over the elements of
the matrix given in (4).

  1. 1.

$a_{1,1},a_{2,2}\in\\{x\in\mathcal{R}_{\theta}:2x=0\\}$ such that $a_{1,1}\neq
a_{2,2}$ and $a_{1,1}\neq a_{2,2}+\lambda$,

  2. 2.

$a_{1,4},a_{2,3}\in\\{x\in\mathcal{R}_{\theta}:3x=x+\lambda\\}$ such that
$a_{1,4}\neq a_{2,3}$ and $a_{1,4}\neq a_{2,3}+\lambda$, and

  3. 3.

$a_{1,2},a_{1,3}\in\\{x\in\mathcal{R}_{\theta}:x\neq 3x,3x=x+\lambda\\}$ such
that $a_{1,2}\neq a_{1,3}$, $a_{1,2}\neq a_{1,3}+\lambda$, $a_{1,2}\neq
3a_{1,3}$, $a_{1,2}\neq 3a_{1,3}+\lambda$.

We construct the following example of a specific Generalized Gau Map $\phi$,
using the abovesaid restrictions, where $\lambda=2+2w$ is considered.

TABLE II: An example of $\phi$: $\mathcal{R}_{\theta}\rightarrow\Sigma_{DNA}^{2}$ is illustrated. Ring element | DNA image | Ring element | DNA image
---|---|---|---
$x$ | $\phi(x)$ | $x$ | $\phi(x)$
$2+2w$ | $AA$ | $w$ | $AC$
$0$ | $TT$ | $3w$ | $CA$
$2w$ | $GG$ | $2+3w$ | $GT$
$2$ | $CC$ | $2+w$ | $TG$
$1$ | $AG$ | $1+w$ | $AT$
$3$ | $GA$ | $3+3w$ | $TA$
$3+2w$ | $CT$ | $3+w$ | $GC$
$1+2w$ | $TC$ | $1+3w$ | $CG$

For a similar map, readers are referred to Table I, [1]. One can observe that,
different Generalized Gau Maps can be constructed by swapping the elements in
the matrix (4), according to the given conditions. This leads us to enumerate
the total number of Generalized Gau map, which can be constructed in this
approach and we have following result.

######  Remark 1.

For a given $\theta\in\mathbb{Z}_{4}+w\mathbb{Z}_{4}$, $2^{11}$ distinct
Generalized Gau mappings
$\phi:\mathcal{R}_{\theta}\rightarrow\Sigma^{2}_{DNA}$ exist such that
$\phi^{-1}(\phi(x)^{r})=3x$ and $\phi^{-1}(\phi(x)^{c})=x+\lambda$ where
$\lambda\in\\{2,2w,2+2w\\}$.

######  Note 1.

In this work, we have considered $\lambda=2+2w$.

Thus motivated, we define the Generalized Gau Distance $d_{G(\theta)}$ over
$\mathcal{R}_{\theta}$ as,

$\begin{array}[]{cc}\mathscr{M}=&\begin{array}[]{cc}&\begin{array}[]{lcccccccccccccccccc}A&&&&G&&&&&&C&&&&&&T&&\end{array}\\\
\begin{array}[]{c}A\\\ G\\\ C\\\
T\end{array}&\left(\begin{array}[]{cccc}a_{1,1}&a_{1,2}&a_{1,3}&a_{1,4}\\\
3a_{1,2}&a_{2,2}&a_{2,3}&3a_{1,3}+2+2w\\\
3a_{1,3}&3a_{2,3}&a_{2,2}+2+2w&3a_{1,2}+2+2w\\\
3a_{1,4}&a_{1,3}+2+2w&a_{1,2}+2+2w&a_{1,1}+2+2w\end{array}\right)\end{array}\end{array}$
(4)

For $x,y\in\mathcal{R}_{\theta}$, from Matrix (4), let $x=a_{i,j}$ and
$y=a_{i^{{}^{\prime}},j^{{}^{\prime}}}$ are the elements of the matrix
$\mathscr{M}$ for some $0\leq i,j\leq 3$ and $0\leq
i^{{}^{\prime}},j^{{}^{\prime}}\leq 3$ then, the Generalized Gau distance
$d_{G(\theta)}$ can be defined as

$d_{G(\theta)}(x,y)=\min\\{1,i+3i^{{}^{\prime}}\\}+\min\\{1,j+3j^{{}^{\prime}}\\},$
(5)

where sum of the indices is calculated over $\mathbb{Z}_{4}$. For any
$\textbf{x}=(x_{1}\ x_{2}\ldots x_{n})$ and $\textbf{y}=(y_{1}\ y_{2}\ldots
y_{n})$ in $\mathcal{R}_{\theta}^{n}$, the Generalized Gau distance
$d_{G(\theta)}(\textbf{x},\textbf{y})=\sum_{i=1}^{n}d_{G(\theta)}(x_{i},y_{i})$
is a metric on $\mathcal{R}_{\theta}^{n}$ induced by the metric on the
elements of the ring $\mathcal{R}_{\theta}$. The same notation $d_{G(\theta)}$
is used for both the metrics on $\mathcal{R}_{\theta}$ and
$\mathcal{R}_{\theta}^{n}$. For a linear code $\mathscr{C}_{\theta}$ on
$\mathcal{R}_{\theta}$, one can define a minimum Generalized Gau distance
$d_{G(\theta)}$ =
$\min\\{d_{G(\theta)}(\textbf{x},\textbf{y}):\textbf{x},\textbf{y}\in\mathscr{C}_{\theta}$
and $\textbf{x}\neq\textbf{y}\\}$.

Using computation, one can verify that, $d_{G(\theta)}$ is a metric on the
ring $\mathcal{R}_{\theta}$ for any $\theta\in\mathcal{R}_{\theta}$. Combining
the Generalized Gau Map $\phi$ and the Generalized Gau Distance
$d_{G(\theta)}$, we have the following theorem.

######  Theorem 2.

The Generalized Gau Map
$\phi:(\mathcal{R}^{n}_{\theta},d_{G(\theta)})\rightarrow(\Sigma^{2n}_{DNA},d_{H})$
is an isometry, $i.e.$, a distance preserving map.

###### Proof.

We will proof the result using Mathematical Induction over the parameter $n$.

Base case: For any given $\theta\in{\mathbb{Z}}_{4}+w{\mathbb{Z}}_{4}$, the
Generalized Gau Map
$\phi:(\mathcal{R}_{\theta},d_{G(\theta)})\rightarrow(\Sigma^{2}_{DNA},d_{H})$
is a distance preserving map, $i.e.$,

$d_{Gau}(x,y)=d_{H}(\phi(x),\phi(y))\mbox{ for any
}x,y\in\mathcal{R}_{\theta}.$

Hypothesis: For given positive integer $m$, assume that the Generalized Gau
Map
$\phi:(\mathcal{R}^{m}_{\theta},d_{G(\theta)})\rightarrow(\Sigma^{2m}_{DNA},d_{H})$
is a distance preserving map, $i.e.$,

$d_{Gau}({\bf x},{\bf y})=d_{H}(\phi({\bf x}),\phi({\bf y}))\mbox{ for any
}{\bf x},{\bf y}\in\mathcal{R}^{m}_{\theta}.$

Inductive step: For any ${\bf x}^{\prime},{\bf
y}^{\prime}\in\mathcal{R}^{m+1}_{\theta}$, consider ${\bf x},{\bf
y}\in\mathcal{R}^{m}_{\theta}$ and $x_{m+1},y_{m+1}\in\mathcal{R}_{\theta}$
such that ${\bf x}^{\prime}$ = $({\bf x}\ x_{m+1})$ and ${\bf y}^{\prime}$ =
$({\bf y}\ y_{m+1})$, where ${\bf x}$ = $(x_{1}\ x_{2}\ \ldots\ x_{m})$ and
${\bf y}$ = $(y_{1}\ y_{2}\ \ldots\ y_{m})$. Then, ${\bf x}^{\prime}$ =
$(x_{1}\ x_{2}\ \ldots\ x_{m}\ x_{m+1})$ and ${\bf y}^{\prime}$ = $(y_{1}\
y_{2}\ \ldots\ y_{m}\ y_{m+1})$. Now,

$\begin{split}d_{Gau}({\bf x}^{\prime},{\bf
y}^{\prime})=&\sum_{i=1}^{m+1}d_{Gau}(x_{i},y_{i})\\\
=&\left(\sum_{i=1}^{m}d_{Gau}(x_{i},y_{i})\right)+d_{Gau}(x_{m+1},y_{m+1})\\\
=&d_{Gau}({\bf x},{\bf y})+d_{Gau}(x_{m+1},y_{m+1})\\\ =&d_{H}(\phi({\bf
x}),\phi({\bf y}))+d_{H}(\phi(x_{m+1}),\phi(y_{m+1}))\\\ =&d_{H}({\bf
x}^{\prime},{\bf y}^{\prime}).\end{split}$

It follows the result. ∎

######  Remark 2.

For any $\textbf{x}\in\mathscr{C}_{\theta}$ over the ring
$\mathcal{R}_{\theta}$,
$\phi^{-1}(\phi(\textbf{x})^{r})\in\mathscr{C}_{\theta}$ if and only if the
DNA code $\phi(\mathscr{C}_{\theta})$ is reversible.

######  Remark 3.

For any $\textbf{x}\in\mathscr{C}_{\theta}$,
$\phi^{-1}(\phi(\textbf{x})^{c})\in\mathscr{C}_{\theta}$ if and only if the
DNA code $\phi(\mathscr{C}_{\theta})$ is complement.

######  Remark 4.

For any $\textbf{x}\in\phi(\mathscr{C}_{\theta})$, if
$\textbf{x}^{r}\in\phi(\mathscr{C}_{\theta})$ and
$\textbf{x}^{c}\in\phi(\mathscr{C}_{\theta})$ then
$\textbf{x}^{rc}\in\phi(\mathscr{C}_{\theta})$.

######  Remark 5.

for any $a$ and $b$ in $\mathcal{R}_{\theta}$, and any ${\bf x}$ = $(x_{1}\
x_{2}\ \ldots\ x_{n})$ and ${\bf y}$ = $(y_{1}\ y_{2}\ \ldots\ y_{n})$ in
$\mathcal{R}^{n}_{\theta}$, consider

$\begin{split}(a{\bf x}+b{\bf y})^{r}=&\left(a(x_{1}\ x_{2}\ \ldots\
x_{n})+b(y_{1}\ y_{2}\ \ldots\ y_{n})\right)^{r}\\\ =&\left((ax_{1}\ ax_{2}\
\ldots\ ax_{n})+(by_{1}\ by_{2}\ \ldots\ by_{n})\right)^{r}\\\
=&(ax_{1}+by_{1}\ ax_{2}+by_{2}\ \ldots\ ax_{n}+by_{n})^{r}\\\
=&(ax_{n}+by_{n}\ ax_{n-1}+by_{n-1}\ \ldots\ ax_{1}+by_{1})\\\ =&(ax_{n}\
ax_{n-1}\ \ldots\ ax_{1})+(by_{n}\ by_{n-1}\ \ldots\ by_{1})\\\ =&(a(x_{n}\
x_{n-1}\ \ldots\ x_{1})+b(y_{n}\ y_{n-1}\ \ldots\ y_{1})\\\ =&a{\bf
x}^{r}+b{\bf y}^{r}.\end{split}$

Using similar approach, one can obtain that, for any positive integer $k$ and
$1\leq i\leq k$, if $\textbf{x}_{i}\in\mathcal{R}^{n}_{\theta}$,
$(\sum_{i=1}^{k}a_{i}\textbf{x}_{i})^{r}=\sum_{i=1}^{k}a_{i}\textbf{x}_{i}^{r}$,
where $a_{i}\in\mathcal{R}_{\theta}$.

######  Lemma 1.

For any positive integer $k$ and $1\leq i\leq k$, if
$\textbf{x}_{i}\in\mathcal{R}^{n}_{\theta}$, then
$\phi^{-1}(\phi(\sum_{i=1}^{k}a_{i}\textbf{x}_{i})^{r})=\sum_{i=1}^{k}a_{i}\phi^{-1}(\phi(\textbf{x}_{i})^{r})$,
where $a_{i}\in\mathcal{R}_{\theta}$.

###### Proof.

For any $x\in\mathcal{R}_{\theta}$, $\phi(x)^{r}$ = $\phi(3x)$, and therefore,
for any ${\bf x}$ = $(x_{1}\ x_{2}\ \ldots\
x_{n})\in\mathcal{R}^{n}_{\theta}$, $\phi({\bf x})^{r}$ = $\phi(3{\bf
x}^{r})$, where ${\bf x}^{r}$ = $(x_{n}\ x_{n-1}\ \ldots\
z_{1})\in\mathcal{R}^{n}_{\theta}$. Now,

$\begin{split}\phi^{-1}\left(\phi\left(\sum_{i=1}^{k}a_{i}\textbf{x}_{i}\right)^{r}\right)&=\phi^{-1}\left(\phi\left(\sum_{i=1}^{k}3a_{i}\textbf{x}_{i}^{r}\right)\right)\\\
&=\sum_{i=1}^{k}3a_{i}\textbf{x}_{i}^{r}\\\
&=\sum_{i=1}^{k}a_{i}(3\textbf{x}_{i}^{r})\\\
&=\sum_{i=1}^{k}a_{i}\phi^{-1}\left(\phi\left(\textbf{x}_{i}\right)^{r}\right)\end{split}$

Hence, it follows the result. ∎

######  Lemma 2.

For a given $\theta$ and for any row ${\bf x}\in G_{\theta}$ over the ring
$\mathcal{R}_{\theta}$, the DNA code $\phi(\langle G_{\theta}\rangle)$
satisfies the reverse constraint if and only if
$\phi^{-1}(\phi(x)^{r})\in\langle G_{\theta}\rangle$, the row span of
$G_{\theta}$ over $\mathcal{R}_{\theta}$.

###### Proof.

Using Remark 2 and Lemma 1, the proof for general case will be similar to the
proof given in Lemma 2, [1], where $\theta=2+2w$ is considered. ∎

######  Lemma 3.

For a given $\theta$ and for any row ${\bf x}\in G_{\theta}$ over the ring
$\mathcal{R}_{\theta}$, the DNA code $\phi(\langle G_{\theta}\rangle)$ is
complement if and only if $\mathbf{2+2w}\in\langle G_{\theta}\rangle$, where
$\mathbf{2+2w}$ is a codeword in $\mathscr{C}_{\theta}$ with each entry
$2+2w$.

For more general case of Lemma 3, we have the following remark.

######  Remark 6.

For a given $\theta$ and for any row ${\bf x}\in G_{\theta}$ over the ring
$\mathcal{R}_{\theta}$, the DNA code $\phi(\langle G_{\theta}\rangle)$ is
complement if and only if $\bm{\lambda}\in\langle G_{\theta}\rangle$, where
$\bm{\lambda}$ is a codeword in $\mathscr{C}_{\theta}$ with each entry
$\lambda\in\\{2,2w,2+2w\\}$.

###### Proof.

Using Remark 3 and Table II, the proof will be similar as given for the
special case for $\theta=2+2w$ in Lemma 3, [1]. ∎

Using Lemma 1, Lemma 2 and Lemma 3, the following theorem gives an insight to
the parameters of the constructed DNA code with specific properties.

######  Remark 7.

For a given $\theta$, suppose $\mathscr{C}_{\theta}(n,M,d_{G(\theta)})$ is a
code over the ring $\mathcal{R}_{\theta}$ with length $n$, the number of the
codewords $M$ and the minimum Gau distance $d_{G(\theta)}$, such that the rows
of the generator matrix of $\mathscr{C}_{\theta}$ satisfies the conditions
given in the Lemma 2 and Lemma 3, then $\phi(\mathscr{C}_{\theta})$ is a
$\mathscr{C}^{\theta}_{DNA}(2n,M,d_{H})$ DNA code with the length $2n$, the
number of the codewords $M$ and the minimum Hamming distance $d_{H}$. The DNA
code $\mathscr{C}^{\theta}_{DNA}$ is reversible and reversible-complement.

Using Remark 7, we propose the following constructions of the families of DNA
codes that satisfy the reverse and reverse-complement constraints from the the
$r^{th}$-order Reed-Muller type codes over the different classes of rings
$\mathcal{R}_{\theta}$.

For integers $m$ and $r$ ($0\leq r\leq m$), the Reed Muller code is denoted as
$\mathcal{R}(r,m)$. In this paper, for integers $m$ and $r$ ($0\leq r\leq m$),
we define the Reed Muller Type code $\mathcal{R}(r,m)$ over the ring
$\mathcal{R}_{\theta}$ for any $\theta\in\mathcal{R}_{\theta}$. The DNA codes
obtained from the Reed Muller Type code satisfies the reverse and reverse-
complement constraints.
For given $\theta$ and a zero divisor $z$ in the ring $\mathcal{R}_{\theta}$,
the $r^{th}$-order Reed Muller Type code $\mathcal{R}(r,m)$ over
$\mathcal{R}_{\theta}$ is given by the generator matrix

$G_{r,m}=\begin{pmatrix}G_{r,m-1}&G_{r,m-1}\\\ 0&G_{r-1,m-1}\\\
\end{pmatrix},\ 1\leq r\leq m-1.$

with

$G_{m,m}\ =\left(\begin{array}[]{c}G_{m-1,m}\\\ 0\ 0\ldots\ 0\
z\end{array}\right)$

and $G_{0,m}\ =\left(1\ 1\ldots 1\right)$ the all one matrix with length
$2^{m}$.

######  Theorem 3.

For the $r^{th}$-order Reed-Muller code $\mathcal{R}(r,m)$ of length $2^{m}$
over the ring $R_{\theta}$ ($\theta\in\\{2,2+2w\\}$), the DNA code
$\mathscr{C}_{DNA}=\phi(\mathcal{R}(r,m))$ satisfies the reverse and reverse-
complement constraints. The $(n,M,d_{H})$ parameters of the DNA code
$\mathscr{C}_{DNA}$ are $n$ = $2^{m+1}$,

$M=\left\\{\,\begin{array}[]{lll}2^{4b-3a}&if&z\in\\{2w\\},\\\
2^{4b-2a}&if&z\in\\{2,2+2w\\},\\\ 2^{4b-a}&if&z\in\\{w,2+w,3w,2+3w\\},\\\
\end{array}\right.\\\ $

and

$d_{H}=\left\\{\,\begin{array}[]{lll}2^{m-r+1}&if&z\in\\{2w,2,2+2w\\},\\\
2^{m-r}&if&z\in\\{w,2+w,3w,2+3w\\},\\\ \end{array}\right.\\\ $

where $a=\sum_{i=0}^{r-1}\binom{m-1}{i}$ and $b=\sum_{i=0}^{r}\binom{m}{i}$.

###### Proof.

The proof follows from the Remark 7, symmetry in generator matrix, and the
following facts.

  * •

The total number of rows of the generator matrix $G_{r,m}$ is
$\sum_{i=0}^{r}\binom{m}{i}$.

  * •

The total number of rows of the generator matrix $G_{r,m}$ that has the
element $z$ is $\sum_{i=0}^{r-1}\binom{m-1}{i}$.

  * •

Any two codewords in the $r^{th}$-order Reed Muller code $\mathcal{R}_{r,m}$
over $\mathcal{R}_{\theta}$ are differ at more then $2^{m-r}-1$ positions.

∎

######  Theorem 4.

For the $r^{th}$-order Reed-Muller code $\mathcal{R}(r,m)$ of length $2^{m}$
over the ring $R_{\theta}$ ($\theta\in\\{1+w,3+w,1+3w,3+3w\\}$), the DNA code
$\mathscr{C}_{DNA}=\phi(\mathcal{R}(r,m))$ satisfy the reverse and reverse-
complement constraints. The $(n,M,d_{H})$ parameters of the DNA code
$\mathscr{C}_{DNA}$ are $n$ = $2^{m+1}$, $M$ = $4^{2b-a}$ and $d_{H}$ =
$2^{m-r+1}$, where $a=\sum_{i=0}^{r-1}\binom{m-1}{i}$ and
$b=\sum_{i=0}^{r}\binom{m}{i}$.

###### Proof.

Similar to Theorem 3, the proof follows from the Remark 7, symmetry in
generator matrix, and the following facts.

  * •

The total number of rows of the generator matrix $G_{r,m}$ is
$\sum_{i=0}^{r}\binom{m}{i}$.

  * •

The total number of rows of the generator matrix $G_{r,m}$ that has the
element $z$ is $\sum_{i=0}^{r-1}\binom{m-1}{i}$.

  * •

Any two codewords in the $r^{th}$-order Reed Muller code $\mathcal{R}_{r,m}$
over $\mathcal{R}_{\theta}$ are differ at more then $2^{m-r}-1$ positions.

∎

######  Theorem 5.

For the $r^{th}$-order Reed-Muller code $\mathcal{R}(r,m)$ of length $2^{m}$
over the ring $R_{\theta}$ ($\theta\in\\{3,1+2w\\}$), the DNA code
$\mathscr{C}_{DNA}=\phi(\mathcal{R}(r,m))$ satisfies the reverse and reverse-
complement constraints. The $(n,M,d_{H})$ parameters of the DNA code
$\mathscr{C}_{DNA}$ are $n$ = $2^{m+1}$,

$M=\left\\{\,\begin{array}[]{lll}2^{4b-3a}&if&z\in\\{2+2w\\},\\\
2^{4b-2a}&if&z\in\\{2,2w\\},\\\ 2^{4b-a}&if&z\in\\{1+w,3+w,1+3w,3+3w\\},\\\
\end{array}\right.\\\ $

and

$d_{H}=\left\\{\,\begin{array}[]{lll}2^{m-r+1}&if&z\in\\{2,2w,2+2w\\},\\\
2^{m-r}&if&z\in\\{1+w,3+w,1+3w,3+3w\\},\\\ \end{array}\right.\\\ $

where $a=\sum_{i=0}^{r-1}\binom{m-1}{i}$ and $b=\sum_{i=0}^{r}\binom{m}{i}$.

###### Proof.

Similar to Theorem 3 and Theorem 4, the proof follows from the Remark 7,
symmetry in generator matrix, and the following facts.

  * •

The total number of rows of the generator matrix $G_{r,m}$ is
$\sum_{i=0}^{r}\binom{m}{i}$.

  * •

The total number of rows of the generator matrix $G_{r,m}$ that has the
element $z$ is $\sum_{i=0}^{r-1}\binom{m-1}{i}$.

  * •

Any two codewords in the $r^{th}$-order Reed Muller code $\mathcal{R}_{r,m}$
over $\mathcal{R}_{\theta}$ are differ at more then $2^{m-r}-1$ positions.

∎

##  IV Dual and Self Dual Codes over the rings $\mathcal{R}_{\theta}$

In this section we study certain specifications of dual codes and establish
sufficient conditions of self dual codes which satisfy the combinatorial
constraints over the ring $\mathcal{R}_{\theta}$.

For the rest of the paper, Standard Inner Product or Euclidean Inner Product
on the ring $\mathcal{R}_{\theta}$ has been considered. For two vectors
$\mathbf{x}=(x_{1}\ x_{2}\ \ldots\ x_{n})$ and $\mathbf{y}=(y_{1}\ y_{2}\
\ldots\ y_{n})$, where $\mathbf{x},\mathbf{y}\in\mathcal{R}^{n}_{\theta}$, the
inner product is defined as
$[\mathbf{x},\mathbf{y}]=\sum\limits_{i=1}^{n}x_{i}y_{i}$ over the ring
$\mathcal{R}_{\theta}$.

The dual code $\mathscr{C}^{\perp}_{\theta}$ of the code
$\mathscr{C}_{\theta}$ over the ring $\mathcal{R}_{\theta}$ is defined as the
set of annihilators of each of the codewords of the code
$\mathscr{C}_{\theta}$,
$i.e.,\mathscr{C}^{\perp}_{\theta}=\\{\mathbf{x}\in\mathcal{R}^{n}_{\theta}:[\mathbf{x},\mathbf{y}]=0,\forall\mathbf{y}\in\mathscr{C}_{\theta}\\}$.
A code $\mathscr{C}_{\theta}$ over the ring $\mathcal{R}_{\theta}$ will be
called self dual, whenever
$\mathscr{C}_{\theta}=\mathscr{C}^{\perp}_{\theta}$.

For the linear codes $\mathscr{C}_{\theta}$ over
$\mathcal{R}_{\theta},\theta\in\\{1+w,3+w,1+3w,3+3w\\}$, $i.e.$, of type
$\\{k_{0},k_{1}\\}$, the dual code $\mathscr{C}^{\perp}_{\theta}$ will be of
type $\\{n-k_{0}-k_{1},k_{1}\\}$. Let $\mathscr{C}_{\theta}$ be a linear code
over $\mathcal{R}_{\theta},\theta\in\\{2,2+2w,3,1+2w\\}$, $i.e.$,
$\mathscr{C}_{\theta}$ is of type $\\{k_{0},k_{1},k_{2},k_{3}\\}$. Then the
dual code $\mathscr{C}^{\perp}_{\theta}$ will be of type
$\\{n-k_{0}-k_{1}-k_{2}-k_{3},k_{3},k_{2},k_{1}\\}$. Hence the following two
remarks are obvious.

######  Remark 8.

If $\mathscr{C}_{\theta}$ is self dual over
$\mathcal{R}_{\theta},\theta\in\\{1+w,3+w,1+3w,3+3w\\}$, then
$2k_{0}+k_{1}=n$.

######  Remark 9.

If $\mathscr{C}_{\theta}$ over
$\mathcal{R}_{\theta},\theta\in\\{2,2+2w,3,1+2w\\}$ is a self dual code, then
$k_{1}=k_{3}$ and $2k_{0}+2k_{1}+k_{2}=n$.

We denote the parity check matrix of $\mathscr{C}_{\theta}$ or the generating
matrix of $\mathscr{C}^{\perp}_{\theta}$ as $G^{\perp}_{\theta}$. Considering
the generating matrix of a linear code $\mathscr{C}_{\theta}$ in standard
form, over the ring $\mathcal{R}_{\theta}$, the generating matrix of the dual
code $\mathscr{C}^{\perp}_{\theta}$ is given by:

  * a)

For the dual code $\mathscr{C}^{\perp}_{\theta}$ over the ring
$\mathcal{R}_{\theta},\theta\in\\{1+w,3+w,1+3w,3+3w\\}$, suppose,
$g_{{}_{1}}=n-k_{0}-k_{1}$. Then,

$G^{\perp}_{\theta}=\left(\begin{array}[]{ccc}C^{T}_{0,1}&C^{T}_{0,2}&I_{g_{{}_{1}}}\\\
-2C^{T}_{1,1}&2I_{k_{1}}&0\\\ \end{array}\right),$ (6)

where $C_{0,1}=(A_{0,1}A_{1,2}-A_{0,2})$ and
$C_{0,2}=(-A_{1,2}),C_{1,1}=(A_{0,1})$.

Note that, $A_{i,j},0\leq i<j\leq 2$ are in the form of the generating matrix
described in (1).

  * b)

For the dual code $\mathscr{C}^{\perp}_{\theta}$ over the ring
$\mathcal{R}_{\theta},\theta\in\\{2,2+2w\\}$, suppose,
$g_{{}_{2}}=n-k_{0}-k_{1}-k_{2}-k_{3}$. Then,

$G^{\perp}_{\theta}=\left(\begin{array}[]{ccccc}C^{T}_{0,1}&C^{T}_{0,2}&C^{T}_{0,3}&C^{T}_{0,4}&I_{g_{{}_{2}}}\\\
-wC^{T}_{1,1}&wC^{T}_{1,2}&-wC^{T}_{1,3}&wI_{k_{3}}&0\\\
2C^{T}_{2,1}&-2C^{T}_{2,2}&2I_{k_{2}}&0&0\\\
-2wC^{T}_{3,1}&2wI_{k_{1}}&0&0&0\\\ \end{array}\right),$ (7)

where,
$C_{0,4}=(-A_{3,4}),C_{0,3}=(A_{2,3}A_{3,4}-A_{2,4}),C_{0,2}=(-A_{1,2}(A_{2,3}A_{3,4}-A_{2,4})+A_{1,3}A_{3,4}-A_{1,4}),C_{0,1}=(A_{0,1}(-C_{0,2})-A_{0,2}(A_{2,3}A_{3,4}-A_{2,4})+A_{0,3}A_{3,4}-A_{0,4})$;
$C_{1,1}=(A_{0,1}(A_{1,2}A_{2,3}-A_{1,3})+A_{0,2}A_{2,3}-A_{0,3}),C_{1,2}=(A_{1,2}A_{2,3}-A_{1,3}),C_{1,3}=(A_{2,3})$;
$C_{2,1}=(A_{0,1}A_{1,2}-A_{0,2}),C_{2,2}=(A_{1,2})$; $C_{3,1}=(A_{0,1})$.

Note that, $A_{i,j},0\leq i<j\leq 4$ are in the form of the generating matrix
described in (2).

  * c)

For the dual code $\mathscr{C}^{\perp}_{\theta}$ over the ring
$\mathcal{R}_{\theta},\theta\in\\{3,1+2w\\}$, suppose,
$g_{{}_{3}}=n-k_{0}-k_{1}-k_{2}-k_{3}$. Then, the corresponding generating
matrix $G^{\perp}_{\theta}$ is given in (8) with additional conditions,

$G^{\perp}_{\theta}=\left(\begin{array}[]{ccccc}C^{T}_{0,1}&C^{T}_{0,2}&C^{T}_{0,3}&C^{T}_{0,4}&I_{g_{{}_{3}}}\\\
-(1+w)C^{T}_{1,1}&(1+w)C^{T}_{1,2}&-(1+w)C^{T}_{1,3}&(1+w)I_{k_{3}}&0\\\
2C^{T}_{2,1}&-2C^{T}_{2,2}&2I_{k_{2}}&0&0\\\
-(2+2w)C^{T}_{3,1}&(2+2w)I_{k_{1}}&0&0&0\\\ \end{array}\right),$ (8)

$C_{0,4}=(-A_{3,4}),C_{0,3}=(A_{2,3}A_{3,4}-A_{2,4}),C_{0,2}=(-A_{1,2}(A_{2,3}A_{3,4}-A_{2,4})+A_{1,3}A_{3,4}-A_{1,4}),C_{0,1}=(A_{0,1}(-C_{0,2})-A_{0,2}(A_{2,3}A_{3,4}-A_{2,4})+A_{0,3}A_{3,4}-A_{0,4})$;
$C_{1,1}=(A_{0,1}(A_{1,2}A_{2,3}-A_{1,3})+A_{0,2}A_{2,3}-A_{0,3}),C_{1,2}=(A_{1,2}A_{2,3}-A_{1,3}),C_{1,3}=(A_{2,3})$;
$C_{2,1}=(A_{0,1}A_{1,2}-A_{0,2}),C_{2,2}=(A_{1,2})$; $C_{3,1}=(A_{0,1})$.

Note that, $A_{i,j},0\leq i<j\leq 4$ are in the form of the generating matrix
described in (3).

The existence of self dual codes of any length has been ensured in [21] for
the Frobenius Rings. Now, we give a construction for all the $16$ rings
$\mathcal{R}_{\theta}$. For the special cases of $\mathcal{R}_{i}$, where
$i=1,2w$, readers are referred to [22].

For any given $\theta$ and any positive integer $n$, note that the code
$\mathscr{C}_{\theta}$ = $\\{(x\ x\ldots x):x\in A\\}$ of length $n$ is a self
dual code, where

$A=\left\\{\begin{array}[]{ll}\mathcal{R}_{\theta}&\mbox{if }4\mid n\\\
\mathcal{Z}_{\theta}&\mbox{if }4\nmid n\mbox{ and }2\mid n\\\
\\{0,2,2w,2+2w\\}&\mbox{otherwise}\end{array}\right.$

and $\mathcal{Z}_{\theta}$ is the set of all zero divisors for the ring
$\mathcal{R}_{\theta}$. Therefore, one can observe the following remark.

######  Remark 10.

There exist self dual codes of all lengths over $\mathcal{R}_{\theta}$.

######  Theorem 6.

For any self-dual code $\mathscr{C}_{\theta}$ of length $n$ over
$\mathcal{R}_{\theta}$

  * i)

at least one of the codewords $\mathbf{2}$, $\mathbf{2w}$ or $\mathbf{2+2w}$
will be present in $\mathscr{C}_{\theta}$ over the ring
$\mathcal{R}_{\theta},\theta\in\\{1+w,3+w,1+3w,3+3w\\}$;

  * ii)

the codeword $\mathbf{2w}$ will be present in $\mathscr{C}_{\theta}$ over the
ring $\mathcal{R}_{\theta},\theta\in\\{2,2+2w\\}$;

  * iii)

the codeword $\mathbf{2+2w}$ will be present in $\mathscr{C}_{\theta}$ over
the ring $\mathcal{R}_{\theta},\theta\in\\{3,1+2w\\}$;

where $\mathbf{x}$ denotes the all $x$ codeword $(x\ x\ldots x)$ for
$x\in\mathcal{R}_{\theta}$.

###### Proof.

Consider the partition of $\mathcal{R}_{\theta}$, for any
$\theta\in\\{1+w,3+w,1+3w,3+3w,2,2+2w,3,1+2w\\}$, into four sets,
$A_{0},A_{1},A_{w^{2}}$ and $A_{(1+w)^{2}}$, where
$A_{i}=\\{x:x\in\mathcal{R},x^{2}=i\\}$. Now for any
$x\in\mathcal{R}_{\theta}$,

$x^{2}=\left\\{\begin{array}[]{ll}0&\mbox{ if }x\in\\{a+wb:a,b\in
2\mathbb{Z}_{4}\\}\\\ 1&\mbox{ if }x\in\\{a+wb:a\in 2\mathbb{Z}_{4}+1,b\in
2\mathbb{Z}_{4}\\}\\\ w^{2}&\mbox{ if }x\in\\{a+wb:a\in 2\mathbb{Z}_{4},b\in
2\mathbb{Z}_{4}+1\\}\\\ 1+2w+w^{2}&\mbox{ if }x\in\\{a+wb:a,b\in
2\mathbb{Z}_{4}+1\\}.\end{array}\right.$

Let $\mathbf{c}=(c_{1}\ c_{2}\ldots c_{n})$ be a codeword in the self dual
code $\mathscr{C}_{\theta}$ of length $n$ over $\mathcal{R}_{\theta}$. Suppose
$n_{x}(\mathbf{c})$ denotes the number of components of a codeword
$\mathbf{c}$ which are from the set $A_{i}$, $i\in\\{0,1,w^{2},(1+w)^{2}\\}$.
Since $\mathscr{C}_{\theta}$ is self dual,
$\langle\mathbf{c},\mathbf{c}\rangle=0,\forall\
\mathbf{c}\in\mathscr{C}_{\theta}$. Hence, $\sum\limits_{i=1}^{n}c_{i}^{2}=0$.
From this we obtain the following equation.

$(n_{1}(\mathbf{c})+n_{(1+w)^{2}}(\mathbf{c}))+w^{2}(n_{w^{2}}(\mathbf{c})+n_{(1+w)^{2}}(\mathbf{c}))+2w(n_{(1+w)^{2}}(\mathbf{c}))=0$
(9)

Now, we classify this approach as following.

  * i)

$\mathcal{R}_{\theta},\theta\in\\{1+w,3+w,1+3w,3+3w\\}=\\{a_{1}+wb_{1}:a_{1},b_{1}\in
2\mathbb{Z}_{4}+1\\}$:
In this case, $w^{2}=a_{1}+wb_{1}$. Then equation (9) becomes,

$\begin{split}(n_{1}(\mathbf{c})+a_{1}(n_{w^{2}}(\mathbf{c}))&+(a_{1}+1)(n_{(1+w)^{2}}(\mathbf{c})))+\\\
&w((b_{1}+2)(n_{(1+w)^{2}}(\mathbf{c}))+b_{1}(n_{w^{2}}(\mathbf{c})))=0.\end{split}$

Hence, we obtain the following two equations,

$n_{1}(\mathbf{c})+a_{1}(n_{w^{2}}(\mathbf{c}))+(a_{1}+1)(n_{(1+w)^{2}}(\mathbf{c}))=0,$
(10) $(b_{1}+2)(n_{(1+w)^{2}}(\mathbf{c}))+b_{1}(n_{w^{2}}(\mathbf{c}))=0.$
(11)

Since $b_{1}$ can be either $1$ or $3$, simplifying equation (11), we get the
immediate result, $n_{(1+w)^{2}}(\mathbf{c})=n_{w^{2}}(\mathbf{c})$ on
$\mathcal{R}_{\theta}$. Using similar approach and the previous result, from
equation (10) we obtain, $n_{1}(\mathbf{c})=n_{w^{2}}(\mathbf{c})$ on
$\mathcal{R}_{\theta}$. Thus

$n_{1}(\mathbf{c})=n_{(1+w)^{2}}(\mathbf{c})=n_{w^{2}}(\mathbf{c})$ (12)

Now, using the multiplication table of $\mathcal{R}_{\theta}$, for any
$x\in\mathcal{R}_{\theta}$, one can observe

$2x=\left\\{\begin{array}[]{ll}0&\mbox{ if }x\in\\{a+wb:a,b\in
2\mathbb{Z}_{4}\\}\\\ 2&\mbox{ if }x\in\\{a+wb:a\in 2\mathbb{Z}_{4}+1,b\in
2\mathbb{Z}_{4}\\}\\\ 2w&\mbox{ if }x\in\\{a+wb:a\in 2\mathbb{Z}_{4},b\in
2\mathbb{Z}_{4}+1\\}\\\ 2+2w&\mbox{ if }x\in\\{a+wb:a,b\in
2\mathbb{Z}_{4}+1\\}\end{array}\right.$

So, the inner product of any codeword $\mathbf{c}=(c_{1}\ c_{2}\ldots c_{n})$
with $\mathbf{2}$ will give,

$\begin{split}\sum_{j=1}^{n}2c_{j}&=0(n_{0}(\mathbf{c}))+2(n_{1}(\mathbf{c}))+2w(n_{w^{2}}(\mathbf{c}))+(2+2w)(n_{(1+w)^{2}}(\mathbf{c}))\\\
&=2(n_{1}(\mathbf{c})+n_{(1+w)^{2}}(\mathbf{c}))+2w(n_{w^{2}}(\mathbf{c})+n_{(1+w)^{2}}(\mathbf{c}))\\\
&=0\hskip 14.22636pt(\mbox{from Equation (\ref{kg eq-1})}).\end{split}$

Therefore, $\mathbf{2}\in\mathscr{C}^{\perp}_{\theta}$, $i.e.$,
$\mathbf{2}\in\mathscr{C}_{\theta}$, since $\mathscr{C}_{\theta}$ is self-
dual. From the chain condition over $\mathcal{R}_{\theta}$, we have,
$\langle\mathbf{2}\rangle=\langle\mathbf{2w}\rangle=\langle\mathbf{2w}\rangle$.
Hence if $\mathscr{C}_{\theta}$ is self-dual, then the code will code will
contain at least one of the codeword $\mathbf{2}$, $\mathbf{2w}$ or
$\mathbf{2+2w}$.

Similar approaches lead to the outcome for the classes of rings
$\mathcal{R}_{\theta},\theta\in\\{2,2+2w\\}$ and $\theta\in\\{3,1+2w\\}$ given
in ii) and iii), respectively and hence, we obtain the desired result. ∎

In the following theorems, the conditions are provided for the construction of
self dual codes considering specific generating matrix over the rings
$\mathcal{R}_{\theta}$ that satisfy the reverse and reverse-complement
constraints.

######  Theorem 7.

Let $\mathscr{C}_{\theta}$ be a self dual code over the ring
$\mathcal{R}_{\theta}$ whose generating matrix is $G_{\theta}=(A_{n}|B_{n})$,
where

$A_{n}=\left(\begin{array}[]{cccc}u&0&\ldots&0\\\ 0&u&\ldots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\ldots&u\\\
\end{array}\right),u\in\mathcal{U}_{\theta}\text{ and
}B_{n}=\left(\begin{array}[]{ccccc}a_{1}&a_{2}&\ldots&a_{n}\\\
a_{n}&a_{1}&\ldots&a_{n-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
a_{2}&a_{3}&\ldots&a_{1}\\\ \end{array}\right),$

a pure circulant matrix of order $n$. Then each $a_{i}$ will also be a unit in
$\mathcal{R}_{\theta}$.

###### Proof.

Since $\mathscr{C}_{\theta}$ is self dual,
$u^{2}+\sum\limits_{i=1}^{n}a_{i}^{2}=0$. Also each row of $G_{\theta}$ will
be orthogonal to each other. Hence,

$2\sum\limits_{j=1}^{n}\sum_{\begin{subarray}{c}k=1\\\
j<k\end{subarray}}^{n}a_{j}\cdot a_{k}=0.$

Without loss of generality if we consider, $u$ is an unit in
$\mathcal{R}_{2+2w}$ then $u^{2}$ will be either $3$ or $1$. If $u^{2}=3$
then,
$\sum\limits_{i=1}^{n}a_{i}^{2}=1\implies(\sum\limits_{i=1}^{n}a_{i}^{2})^{2}=\sum\limits_{i=1}^{n}a_{i}^{2}=1$.
Now there are exactly $4$ solutions of this equation $x^{2}=1$ in
$\mathcal{R}_{2+2w}$, i.e., $a_{i}\in\\{1,3,1+2w,3+2w\\}$. The same result can
be obtained by similar argument, for $u^{2}=1$ in $\mathcal{R}_{\theta}$.
Proofs for other rings $\mathcal{R}_{\theta}$,
$\theta\in\mathbb{Z}_{4}+w\mathbb{Z}_{4}\setminus\\{2+2w\\}$ will be the same.
Hence, we obtain the desired result. ∎

######  Theorem 8.

The generating matrix $G_{\theta}$ of a self dual code $\mathscr{C}_{\theta}$,
considered in Theorem 7, will be closed under reverse constraint if
$a_{i}\in\mathcal{U}_{\theta}\ \forall i\in\\{1,2,\ldots n\\}$.

###### Proof.

Recall the generating matrix $G_{\theta}$ of $\mathscr{C}_{\theta}$ in Theorem
7. Now the reverse of the first row of $G_{\theta}$ will be $(a_{n}\
a_{n-1}\ldots a_{1}\ldots 0\ldots u)$. By inspection we can say that,
$a_{n}=\alpha\cdot u$, where $\alpha\in\mathcal{R}_{\theta}$. So we obtain,
$a_{i}=\alpha_{i}u,\ i=1,2,\ldots n$. Now each $\alpha_{i}$ will be a unit in
$\mathcal{R}_{\theta}$, as $a_{i}$ and $u$ are units in $\mathcal{R}$.
Therefore, $a_{i}=u_{i}$, where $u_{i}=\alpha_{i}u,\ i=1,2,\ldots n$ and hence
the result follows. ∎

Using the same argument, as applied in Theorem 8 the following theorem is
obvious.

######  Theorem 9.

The generating matrix $G_{\theta}$ of a self dual code $\mathscr{C}_{\theta}$,
considered in Theorem 7, will be closed under reverse-complement constraint,
if $a_{i}\in 3\mathcal{U}_{\theta}\ \forall i\in\\{1,2,\ldots n\\}$.

###### Proof.

The proof is straightforward. ∎

######  Remark 11.

As $3\in\mathcal{U}_{\theta}$ in $\mathcal{R}_{\theta}$, $G_{\theta}$ will be
closed under reverse-complement constraint if $a_{i}\in\mathcal{U}_{\theta}\
\forall i\in\\{1,2,\ldots n\\}$.

##  V Bounds for the codes over the ring $\mathcal{R}_{\theta}$

In this section we obtain Sphere Packing-like bound, Gilbert Vershamov-like
bound, Singleton-like bound and Plotkin-like bound for the codes over the ring
$\mathcal{R}_{\theta}$.

###  V-A Sphere Packing-like Bound on the codes over $\mathcal{R}_{\theta}$

In order to establish Sphere Packing-like bound, first we have defined the
sphere in Definition 1 and the establish the Lemma 4 and Lemma 14.

######  Definition 1.

For any vector $\mathbf{c}$ in $\mathcal{R}^{n}_{\theta}$ and any integer
$r\geq 0$, the sphere of radius $r$ and center $\mathbf{c}$, denoted as
$S_{r}(\mathbf{c})$, is the set
$\\{\mathbf{x}\in\mathcal{R}^{n}_{\theta}:d_{G(\theta)}(\mathbf{c},\mathbf{x})\leq
r\\}$.

For any $\textbf{c}\in\mathcal{R}^{n}_{\theta}$, if the set
$C_{r}(\mathbf{c})=\\{\mathbf{x}\in\mathcal{R}^{n}_{\theta}:d_{G(\theta)}(\mathbf{c},\mathbf{x})=r\\}$,
then

$|S_{r}(\textbf{c})|=\sum_{i=0}^{r}|C_{i}(\textbf{c})|.$ (13)

######  Lemma 4.

Let $\mathscr{C}_{\theta}(n,M,d_{G(\theta)})$ be a code over the ring
$\mathcal{R}_{\theta}$. Then $S_{r}(\mathbf{c}_{1})\cap
S_{r}(\mathbf{c}_{2})=\emptyset$, when
$r\leq\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor,\ \forall\
\mathbf{c}_{1},\mathbf{c}_{2}\in\mathscr{C}_{\theta}$.

###### Proof.

Consider $S_{r}(\mathbf{c}_{1})$ and $S_{r}(\mathbf{c}_{2})$ for two distinct
codewords $\mathbf{c}_{1},\mathbf{c}_{2}\in\mathscr{C}_{\theta}$. For a vector
$\mathbf{x}$ is in both $S_{r}(\mathbf{c}_{1})$ and $S_{r}(\mathbf{c}_{2})$,
from the triangle inequality we obtain,
$d_{G(\theta)}(\mathbf{c}_{1},\mathbf{c}_{2})\leq
d_{G(\theta)}(\mathbf{c}_{1},\mathbf{x})+d_{G(\theta)}(\mathbf{x},\mathbf{c}_{2})\leq
r+r=2r$, a contradiction to $d_{G(\theta)}\geq 2r+1$. Hence,
$S_{r}(\mathbf{c}_{1})\cap S_{r}(\mathbf{c}_{2})=\emptyset$, whenever
$r\leq\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor,\ \forall\
\mathbf{c}_{1},\mathbf{c}_{2}\in\mathscr{C}_{\theta}$. ∎

######  Lemma 5.

For $\textbf{c}\in\mathcal{R}^{n}_{\theta}$ and a non-negative integer $r$
$(\leq 2n)$, if
$C_{r}(\textbf{c})=\\{\textbf{x}\in\mathcal{R}^{n}_{\theta}:d_{G(\theta)}(\textbf{c},\textbf{x})=r\\}$,
then,

$|C_{r}(\textbf{c})|=\sum\limits_{i=\max\\{0,r-n\\}}^{\left\lfloor\frac{r}{2}\right\rfloor}\frac{n!}{i!(r-2i)!(n-r+i)!}9^{i}6^{r-2i}.$
(14)

###### Proof.

Let $\mathbf{c}$ be a fixed vector in $\mathcal{R}_{\theta}$. We will consider
the number of vectors $\mathbf{x}$ that have Gau distance exactly $r$ from
$\mathbf{c}$, where $r\leq 2n$. From the matrix $\mathscr{M}$ given in (4), we
can see that the Gau distance between a given element and any distinct element
in the same row or column is one, and the Gau distance between the given
element and the remaining distinct elements is two. Suppose for ${\bf x}\in
C_{r}(\textbf{c})$, there are $i$ distinct positions where the elements at
those positions at ${\bf x}$ and c are two. Thus, there are $r-2i$ distinct
positions other than the $i$ positions where the elements at those positions
at ${\bf x}$ and c are one, and elements at remaining postilions $n-r+i$ (=
$n-((r-2i)+i)$) are the same. So, for $i$ elements, which are differ by Gau
distance $2$, can be chosen in $9^{i}$ ways and the other $r-2i$ elements can
be chosen in $6^{r-2i}$ ways. Now among all these $n$ elements, $i$ and $r-2i$
elements can be arranged in $i!$ and $(r-2i)!$ ways, respectively, among $n$
places. The remaining coordinates can be arranged in $(n-r+i)!$ ways. So, the
total number of vectors in $C_{r}(\mathbf{c})$ will be,

$\sum\limits_{i=\max\\{0,r-n\\}}^{\left\lfloor\frac{r}{2}\right\rfloor}\frac{n!}{i!(r-2i)!(n-r+i)!}9^{i}6^{r-2i}.$

∎

######  Theorem 10 (Sphere Packing-like Bound).

A code $\mathscr{C}_{\theta}(n,M,d_{G(\theta)})$ over the ring
$\mathcal{R}_{\theta}$ satisfies,

$M\sum_{r=0}^{\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor}\sum_{i=0}^{\left\lfloor\frac{r}{2}\right\rfloor}\frac{n!9^{i}6^{r-2i}}{i!(r-2i)!(n-r+i)!}\leq
16^{n}.$

###### Proof.

From Lemma 4, any two spheres of radius
$m=\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor$ centred on distinct
codewords will be disjoint. So, from the Inclusion–Exclusion principle,

$\left|\bigcup_{\textbf{c}\in\mathscr{C}_{\theta}}S_{m}(\textbf{c})\right|=\sum_{\textbf{c}\in\mathscr{C}_{\theta}}|S_{m}(\textbf{c})|.$

Therefore, the total number of codewords in each of the $M$ spheres of radius
$m$ centred on the $M$ codewords is bounded by $16^{n}$, $i.e.$

$16^{n}\geq\left|\bigcup_{\textbf{c}\in\mathscr{C}_{\theta}}S_{m}(\textbf{c})\right|=\sum_{\textbf{c}\in\mathscr{C}_{\theta}}|S_{m}(\textbf{c})|=\sum_{\textbf{c}\in\mathscr{C}_{\theta}}\sum_{r=0}^{\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor}|C_{r}(\textbf{c})|.$

But for given non-negative integer $r\ (\leq 2n)$, $|C_{r}(\textbf{c})|$ is
same for each $\textbf{c}\in\mathscr{C}_{\theta}$. Therefore,

$\begin{split}16^{n}\geq
M\sum_{r=0}^{\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor}|C_{r}(\textbf{c})|.\end{split}$

Hence, from Lemma 14 and the fact that, for $0\leq
r\leq\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor$, $max\\{0,r-n\\}=0$,
we obtain,

$16^{n}\geq
M\sum\limits_{r=0}^{\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor}\sum_{i=0}^{\left\lfloor\frac{r}{2}\right\rfloor}\frac{n!9^{i}6^{r-2i}}{i!(r-2i)!(n-r+i)!}$

∎

######  Corollary 11.

For the linear code $\mathscr{C}_{\theta}$ of type
$\\{k_{0},k_{1},k_{2},k_{3}\\}$ over $\mathcal{R}_{\theta}$ for
$\theta\in\\{2,3,1+2w,2+2w\\}$, the Sphere Packing-like bound will be given
by,

$\sum\limits_{r=0}^{\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor}\sum_{i=0}^{\left\lfloor\frac{r}{2}\right\rfloor}\frac{n!9^{i}6^{r-2i}}{i!(r-2i)!(n-r+i)!}\leq
16^{n-\left(k_{0}+\frac{3k_{1}}{4}+\frac{k_{2}}{2}+\frac{k_{3}}{4}\right)}.$

######  Corollary 12.

For the linear code $\mathscr{C}_{\theta}$ of type $\\{k_{0},k_{1}\\}$ over
$\mathcal{R}_{\theta}$ for $\theta\in\\{1+w,1+3w,3+w,3+3w\\}$, the Sphere
Packing-like bound will be given by,

$\sum\limits_{r=0}^{\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor}\sum_{i=0}^{\left\lfloor\frac{r}{2}\right\rfloor}\frac{n!9^{i}6^{r-2i}}{i!(r-2i)!(n-r+i)!}\leq
16^{n-\left(k_{0}+\frac{k_{1}}{2}\right)}.$

###  V-B Gilbert Varshamov-like Bound

######  Theorem 13 (Gilbert Varshamov-like Bound).

If $A_{16}(n,d_{G(\theta)})$ is the maximum possible size of a code
$\mathscr{C}_{\theta}$ with length $n$ and minimum Gau distance
$d_{G(\theta)}$ over the ring $\mathcal{R}_{\theta}$ then

$16^{n}\leq
A_{16}(n,d_{G(\theta)})\sum_{r=0}^{d_{G(\theta)}-1}\sum_{i=\max\\{0,r-n\\}}^{\left\lfloor\frac{r}{2}\right\rfloor}\frac{n!9^{i}6^{r-2i}}{i!(r-2i)!(n-r+i)!}.$

###### Proof.

Let $\mathscr{C}_{\theta}$ $(n,M,d_{G(\theta)})$ be a code over the ring
$\mathcal{R}_{\theta}$ with maximal size $|\mathscr{C}_{\theta}|$ = $M$ =
$A_{16}(n,d_{G(\theta)})$. Therefore, for each
$\textbf{x}\in\mathcal{R}^{n}_{\theta}$, there exists at least one codeword
$\textbf{c}\in\mathscr{C}_{\theta}$ such that
$d_{G(\theta)}(\textbf{c},\textbf{x})\leq d_{G(\theta)}-1$. So,

$\mathcal{R}^{n}_{\theta}=\bigcup_{\textbf{c}\in\mathscr{C}_{\theta}}S_{d_{G(\theta)}-1}(\textbf{c}),$

where the sphere $S_{d_{G(\theta)}-1}(\textbf{c})$ is given by Definition 1.
But from Lemma 14 and Equation 13, for each
$\textbf{c}\in\mathscr{C}_{\theta}$ the sphere size will be

$\begin{split}|S_{d_{G(\theta)}-1}(\textbf{c})|=&\sum_{r=0}^{d_{G(\theta)}-1}|C_{r}(\textbf{c})|\\\
=&\sum_{r=0}^{d_{G(\theta)}-1}\sum_{i=\max\\{0,r-n\\}}^{\left\lfloor\frac{r}{2}\right\rfloor}\frac{n!9^{i}6^{r-2i}}{i!(r-2i)!(n-r+i)!}.\end{split}$

Hence,

$\begin{split}16^{n}=&|\mathcal{R}^{n}_{\theta}|\\\
=&\left|\bigcup_{\textbf{c}\in\mathscr{C}_{\theta}}S_{d_{G(\theta)}-1}(\textbf{c})\right|\\\
\leq&\sum_{\textbf{c}\in\mathscr{C}_{\theta}}\sum_{r=0}^{d_{G(\theta)}-1}\sum_{i=\max\\{0,r-n\\}}^{\left\lfloor\frac{r}{2}\right\rfloor}\frac{n!9^{i}6^{r-2i}}{i!(r-2i)!(n-r+i)!}\\\
=&A_{16}(n,d_{G(\theta)})\sum_{r=0}^{d_{G(\theta)}-1}\sum_{i=\max\\{0,r-n\\}}^{\left\lfloor\frac{r}{2}\right\rfloor}\frac{n!9^{i}6^{r-2i}}{i!(r-2i)!(n-r+i)!}.\end{split}$

∎

###  V-C Singleton-like Bound

To establish the Singleton-like Bound, we first need to prove the following
Lemma.

######  Lemma 6.

Let $\mathscr{C}_{\theta}$ $(n,M,d_{G(\theta)})$ be a code over the ring
$\mathcal{R}_{\theta}$. Then by deleting the first
$\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor\ (=r)$ letters in every
codeword of $\mathscr{C}_{\theta}$, we will obtain exactly $M$ distinct
codewords of length $n-\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor$,
i.e. $n-r$.

###### Proof.

Let $\mathbf{x}=(x_{1}\ x_{2}\ldots\ x_{r-1}\ x_{r}\ x_{r+1}\ldots\ x_{n})$
and $\mathbf{y}=(y_{1}\ y_{2}\ldots\ y_{r-1}\ y_{r}$ $y_{r+1}\ldots\ y_{n})$
be distinct codewords in $\mathscr{C}_{\theta}$, such that $(x_{r+1}\ldots\
x_{n})$ = $(y_{r+1}\ldots$ $y_{n})$. Then $\exists$ at least one
$j\in\\{1,2,\ldots,r\\}$ such that $d_{G(\theta)}(x_{j},y_{j})\neq 0$ (since,
$\mathbf{x}\neq\mathbf{y}$).
Now consider the general case,
$d_{G(\theta)}(\mathbf{x},\mathbf{y})=r_{1}+2r_{2}$, where $r=r_{1}+r_{2}$ and
$r_{1},r_{2}$ are nonnegative integers. Then, $d_{G(\theta)}\leq
r_{1}+2r_{2}$, $i.e.$,
$\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor\leq\left\lfloor\frac{(r_{1}+2r_{2})-1}{2}\right\rfloor$
which contradicts,
$r=\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor<\left\lfloor\frac{2r-1}{2}\right\rfloor<r$.
Hence there will be exactly $M$ distinct codewords of length $n-r$ in
$\mathscr{C}_{\theta}$. ∎

######  Remark 12.

From Lemma 6, it is obvious that $\exists$ at least one $l\geq r+1$ such that
$d_{G(\theta)}(x_{l},y_{l})\neq 0$.

Using the above lemma we can proceed to the Singleton-like bound for a code
over $\mathcal{R}_{\theta}$.

######  Theorem 14 (Singleton-like Bound).

If $\mathscr{C}_{\theta}(n,M,d_{G(\theta)})$ is a code (not necessarily
linear) over $\mathcal{R}_{\theta}$, then $M\leq
16^{n-\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor}.$

###### Proof.

From Lemma 6, we have exactly $M$ distinct codewords of length $n-r$, by
deleting the first $r\ (=\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor)$
letters in every codewords of $\mathcal{C}_{\theta}$. The order of the ring
$\mathcal{R}_{\theta}$ is sixteen, so $M\leq
16^{n-\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor}$ which is the
Singleton-like bound for any code $\mathcal{C}_{\theta}$ (not necessarily
linear) over $\mathcal{R}_{\theta}$. ∎

######  Corollary 15.

If the code $\mathscr{C}_{\theta}(n,M,d_{G(\theta)})$ is linear and of type
$\\{k_{0},k_{1},k_{2},k_{3}\\}$ over $\mathcal{R}_{\theta}$ for
$\theta\in\\{2,3,1+2w,2+2w\\}$, then

${\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor}\leq
n-\left(k_{0}+\frac{3k_{1}}{4}+\frac{k_{2}}{2}+\frac{k_{3}}{4}\right).$

######  Corollary 16.

If the code $\mathscr{C}_{\theta}(n,M,d_{G(\theta)})$ is linear and of type
$\\{k_{0},k_{1}\\}$ over $\mathcal{R}_{\theta}$ for
$\theta\in\\{1+w,3+w,1+3w,3+3w\\}$, then

${\left\lfloor\frac{d_{G(\theta)}-1}{2}\right\rfloor}\leq
n-\left(k_{0}+\frac{k_{1}}{2}\right).$

######  Definition 2.

Any code $\mathscr{C}_{\theta}$ defined on the ring $\mathcal{R}_{\theta}$ is
called Maximum Gau Distance Separable (MGDS) code, if the code
$\mathscr{C}_{\theta}$ satisfies the Singleton-like bound as given in Theorem
14.

######  Remark 13.

For any positive integer $n$ and $\theta\in\mathbb{Z}_{4}+w\mathbb{Z}_{4}$,
the code $\mathscr{C}_{\theta}$ = $\mathcal{R}^{n}_{\theta}$ with the
parameter $(n,16^{n},1)$ is MGDS code over the ring $\mathcal{R}_{\theta}$.

######  Remark 14.

For any positive integer $n$, if $\textbf{a}\in\\{2,2+2w\\}^{n}$ then the code
$\mathscr{C}_{\theta}$ = $\langle\textbf{a}\rangle$ with the parameter
$(n,4,2n)$ is MGDS code over the ring $\mathcal{R}_{\theta}$, where
$\theta\in\\{2,2+2w\\}$.

######  Remark 15.

For any positive integer $n$, if $\textbf{a}\in\\{2,2w\\}^{n}$ then the code
$\mathscr{C}_{\theta}$ = $\langle\textbf{a}\rangle$ with the parameter
$(n,4,2n)$ is MGDS code the ring $\mathcal{R}_{\theta}$, where
$\theta\in\\{3,1+2w\\}$.

######  Remark 16.

For any positive integer $n$, if $\textbf{a}\in\\{2,2w,2+2w\\}^{n}$ then the
code $\mathscr{C}_{\theta}$ = $\langle\textbf{a}\rangle$ with the parameter
$(n,4,2n)$ is MGDS code the ring $\mathcal{R}_{\theta}$, where
$\theta\in\\{1+w,3+w,1+2w,3+3w\\}$.

###  V-D Plotkin-like bound on the codes over $\mathcal{R}_{\theta}$

In the following we establish the Plotkin-like bound considering the
Generalized Gau distance over the rings $\mathcal{R}_{\theta}$.

######  Theorem 17 (Plotkin-like Bound).

A code $\mathscr{C}_{\theta}$ $(n,M,d_{G(\theta)})$ over the ring
$\mathcal{R}_{\theta}$ will satisfy,

$M\leq\left\lfloor\frac{2d_{G(\theta)}}{2d_{G(\theta)}-3n}\right\rfloor\
\mbox{ for }2d_{G(\theta)}>3n.$

###### Proof.

For positive integers $n$ and $M$, and $i=1,2,\ldots,M$, consider
$\mathbf{x}_{i}=(x_{i,1}\ x_{i,2}\ldots x_{i,n})\in\mathscr{C}_{\theta}$. We
will prove the theorem by using lower and upper bounds of
$\sum_{i=1}^{M}\sum_{j=1}^{M}d_{G(\theta)}(\mathbf{x}_{i},\mathbf{x}_{j}).$
For lower bound, there are $M$ choices for $\mathbf{x}_{i}$ and hence, $M-1$
choices are left for $\mathbf{x}_{j}$. Therefore,

$\sum_{i=1}^{M}\sum_{\begin{subarray}{c}j=1\\\ i\neq
j\end{subarray}}^{M}d_{G(\theta)}(\mathbf{x}_{i},\mathbf{x}_{j})\geq
M(M-1)d_{G(\theta)}.$ (15)

For upper bound, consider a matrix of order $M\times n$, whose rows represent
the codewords of $\mathscr{C}_{\theta}$. So,

$\sum_{i=1}^{M}\sum_{\begin{subarray}{c}j=1\\\ i\neq
j\end{subarray}}^{M}d_{G(\theta)}(\mathbf{x}_{i},\mathbf{x}_{j})=\sum_{l=1}^{n}\sum_{i=1}^{M}\sum_{\begin{subarray}{c}j=1\\\
i\neq j\end{subarray}}^{M}d_{G(\theta)}(x_{i,l},x_{j,l}).$ (16)

For each $m\in\mathcal{R}_{\theta}$, let $s_{m,l}$ ($l=1,2,\ldots n$) denotes
the occurrence of the ring element $m$ in the $l^{th}$ column of the matrix.
Then for $l=1,2,\ldots,n$,

$\sum_{m\in\mathcal{R}}s_{m,l}=M.$ (17)

Let us also define two sets
$A_{m}=\\{y:d_{G}(m,y)=1,y\in\mathcal{R}_{\theta}\\}$ and
$B_{m}=\\{y:d_{G}(m,y)=2,y\in\mathcal{R}_{\theta}\\}$, for each
$m\in\mathcal{R}_{\theta}$, using properties of Gau distance. Then $A_{m}\cup
B_{m}\cup\\{m\\}=\mathcal{R}_{\theta}$, for each $m$. Now, for
$l=1,2,\ldots,n$, if $s_{i,l}$ is the number of occurrence of the element
$i\in\mathcal{R}_{\theta}$ in the column $l$ of the matrix then

$\displaystyle\sum_{i=1}^{M}\sum_{\begin{subarray}{c}j=1\\\ i\neq
j\end{subarray}}^{M}d_{G}(x_{i,l},x_{j,l})$
$\displaystyle=\sum_{m\in\mathcal{R}_{\theta}}\binom{s_{m,l}}{1}\left(1\sum_{j\in
A_{m}}\binom{s_{j,l}}{1}+2\sum_{j\in B_{m}}\binom{s_{j,l}}{1}\right)$ (18)
$\displaystyle=\sum_{m\in\mathcal{R}_{\theta}}s_{m,l}\left(\sum_{j\in
A_{m}}s_{j,l}+2\sum_{j\in B_{m}}s_{j,l}\right).$ (19)

The symmetricity of the RHS in Equation (18) implies that it will be maximum
along with the constraint Equation (17) if $s_{i,l}=M/16$ for each
$i\in\mathcal{R}_{\theta}$ and $l=1,2,\ldots,n$. Hence,

$\sum_{m\in\mathcal{R}_{\theta}}s_{m,l}\left(\sum_{j\in
A_{m}}s_{j,l}\right.+\left.2\sum_{j\in
B_{m}}s_{j,l}\right)\leq\sum_{m\in\mathcal{R}_{\theta}}\frac{24}{16^{2}}M^{2}=\frac{3M^{2}}{2},$
(20)

where $|A_{m}|=6$ and $|B_{m}|=9$ for each $m\in\mathcal{R}_{\theta}$. From
Inequality (15), Equation (16), Equation (18) and Inequality (20), we obtain,
$M(M-1)d_{G(\theta)}\leq\sum_{l=1}^{n}\frac{3M^{2}}{2}.$ Therefore,
$M\leq\frac{2d_{G(\theta)}}{2d_{G(\theta)}-3n}.$ Note that $M,n$ and
$d_{G(\theta)}$ are finite positive integers. Therefore,
$M\leq\left\lfloor\frac{2d_{G(\theta)}}{2d_{G(\theta)}-3n}\right\rfloor\mbox{
when }2d_{G(\theta)}>3n.$ ∎

######  Corollary 18.

If the code $\mathscr{C}_{\theta}(n,M,d_{G(\theta)})$ is linear and of type
$\\{k_{0},k_{1},k_{2},k_{3}\\}$ over $\mathcal{R}_{\theta}$,
$\theta\in\\{2,3,1+2w,2+2w\\}$, then,

$\left(k_{0}+\frac{3k_{1}}{4}+\frac{k_{2}}{2}+\frac{k_{3}}{4}\right)\leq
log_{16}\left(\frac{2d_{G(\theta)}}{2d_{G(\theta)}-3n}\right)$

when $2d_{G(\theta)}>3n$.

######  Corollary 19.

If the code $\mathscr{C}_{\theta}(n,M,d_{G(\theta)})$ is linear and of type
$\\{k_{0},k_{1}\\}$ over $\mathcal{R}_{\theta}$,
$\theta\in\\{1+w,1+3w,3+w,3+3w\\}$, then for $2d_{G(\theta)}>3n$,

$\left(k_{0}+\frac{k_{1}}{2}\right)\leq\log_{16}\left(\frac{2d_{G(\theta)}}{2d_{G(\theta)}-3n}\right).$

######  Remark 17.

For any positive integer $n$, if $\textbf{a}\in\\{2,2+2w\\}^{n}$ then the
family of codes $\mathscr{C}_{\theta}(n,4,2n)$ = $\langle\textbf{a}\rangle$
satisfies the Plotkin-like bound with equality over the ring
$\mathcal{R}_{\theta}$, where $\theta\in\\{2,2+2w\\}$.

######  Remark 18.

For any positive integer $n$, if $\textbf{a}\in\\{2,2w\\}^{n}$ then the class
of codes $\mathscr{C}_{\theta}(n,4,2n)$ = $\langle\textbf{a}\rangle$ will be
optimal with respect to the Plotkin-like bound over the ring
$\mathcal{R}_{\theta}$, where $\theta\in\\{3,1+2w\\}$.

######  Remark 19.

For any positive integer $n$, if $\textbf{a}\in\\{2,2w,2+2w\\}^{n}$ then the
class of codes $\mathscr{C}_{\theta}(n,4,2n)$ = $\langle\textbf{a}\rangle$
satisfies the Plotkin-like bound optimally over the ring
$\mathcal{R}_{\theta}$, where $\theta\in\\{1+w,3+w,1+2w,3+3w\\}$.

##  VI Conclusion

The present paper includes the exhaustive study of the $16$ ring structures
$\mathbb{Z}_{4}+w\mathbb{Z}_{4}$, $w^{2}\in\mathbb{Z}_{4}+w\mathbb{Z}_{4}$. We
extended the concept of Gau map and Gau distance, proposed in [1], over the
entire class of $16$ rings. The isometry between the codes over the rings and
the analogous DNA codes has been established universally considering Gau
distance and Hamming distance over the elements of the rings and DNA strings
of length $2$, respectively. In addition to this, necessary and sufficient
conditions have been proposed so that the constructed DNA codes satisfy
reverse and reverse-complement constraints. We enrich the study of dual and
self dual codes along with the characterization of corresponding generator
matrix with different specifications in a general manner over the rings.
Prominent bounds $i.e.$, Sphere Packing-like bound, GV-like bound, Singleton-
like bound and Plotkin-like bound have been proposed considering the Gau
distance over all the rings. Parameters for special classes of optimal codes
have been proposed that meet Singleton-like bound and Plotkin-like bound,
respectively. Also the construction of a class of DNA codes using the Reed-
Muller type codes over the rings has been given, which satisfies reverse and
reverse-complement constraints. In future, it will be an interesting task to
define Gau weight over the rings considered in this work. Utilizing the
advantage, the derivation of McWilliam’s Identity can be done in future. The
construction of different families of optimal codes for each of the proposed
bounds will also be an engaging work.

## Acknowledgment

This work is funded by the Deanship of Scientific Research (DSR), King
Abdulaziz University, Jeddah under grant number (DF-594-130-1441). The
authors, therefore, gratefully acknowledge DSR technical and financial
support.

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