# On DNA Codes Over the Non-Chain Ring
$\mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4}$ with $u^{3}=1$

Shibsankar Das, Krishna Gopal Benerjee and Adrish Banerjee  The Department of
Electrical Engineering
Indian Institute of Technology Kanpur, UP, India
Email: {shibsankar, kgopal<EMAIL_ADDRESS>
###### Abstract

In this paper, we present a novel design strategy of DNA codes with length
$3n$ over the non-chain ring
$R=\mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4}$ with $64$ elements and
$u^{3}=1$, where $n$ denotes the length of a code over $R$. We first study and
analyze a distance conserving map defined over the ring $R$ into the
length-$3$ DNA sequences. Then, we derive some conditions on the generator
matrix of a linear code over $R$, which leads to a DNA code with reversible,
reversible-complement, homopolymer $2$-run-length, and $\frac{w}{3n}$-GC-
content constraints for integer $w$ ($0\leq w\leq 3n$). Finally, we propose a
new construction of DNA codes using Reed-Muller type generator matrices. This
allows us to obtain DNA codes with reversible, reversible-complement,
homopolymer $2$-run-length, and $\frac{2}{3}$-GC-content constraints.

###### Index Terms:

DNA Nucleotides, Reversible, Reversible-Complement, Homopolymer Run-Length
Free, and GC-Content

##  I Introduction

In recent years, the explosion of social networking has produced an ever-
increasing amount of digital data [1]. By $2025$, it is expected that $491$
exabytes (EB) of data per day will be accumulated around the world [2]. Due to
this rapid data growth, storing such a massive amount of data with higher and
efficient storage capacity becomes essential. Thus, it is important to
investigate a new medium to store the data effectively. Owing to its higher
storage capacity, lower energy consumption [3], and outstanding storable time
[4], the synthetic deoxyribonucleic acid (DNA) is an ideal medium to store a
massive amount of digital data [5]. The DNA-based storage is extremely dense
with a limit of $10^{9}GB/mm^{3}$, and it has a long storable capability with
a half-life of more than $500$ years [4].

The DNA strands can be considered as sequences of four nucleotides: A
(Adenine), T (Thymine), G (Guanine), and C (Cytosine). The Watson-Crick
complement (or, simply complement) of DNA nucleotide is given by
$\text{C}^{c}=$ G, $\text{G}^{c}=$ C, $\text{A}^{c}=$ T and $\text{T}^{c}=$ A.
In DNA data storage, the digital data is translated into DNA sequences
(constituted by four symbols A, T, G, and C). For example, two binary bits are
mapped into a single nucleotide [6]. That is, $00\rightarrow$ A,
$10\rightarrow$ G, $01\rightarrow$ C and $11\rightarrow$ T. To extract the
original data, the DNA strings are sequenced and synthesized inversely into
the binary data.

In [7], it has been shown that two biochemical constraints, homopolymer run-
length constraint and the fixed GC-content of DNA sequences, can be utilized
to combat the sequencing and synthesizing errors in the retrieval process. The
fixed GC-content constraint ensures that all DNA codewords enjoy a similar
thermodynamic characteristic and the homopolymer run-length constraint limits
the maximum number of consecutive repetitions of nucleotides in the encoded
DNA [8]-[10].

In the oligonucleotide (i.e., single-stranded DNA) library, the non-specific
DNA hybridization is a leading source of errors [11]. To combat this unwanted
DNA hybridization (or to avoid the errors in computation), the minimum Hamming
distance between distinct DNA codewords, the reversible, and reversible-
complement DNA codewords has been studied extensively in [11, 12, 13, 14, 15,
16, 17]. Thus, it is more desirable that the oligonucleotide library should
have a greater number of DNA codewords satisfying the maximum number of
constraints for the given Hamming distance $d_{H}$.

In the design of DNA sequences from the ring structure, sequences are
constructed by properly identifying a mapping from the ring into the set of
DNA nucleotides [18]. The major advantage of using an algebraic approach over
the computational methods is that it can produce an optimal DNA code set with
a relatively longer sequence length [12]. To date, there have been many
algebraic construction methods for DNA sequences with certain constraints [19,
20, 21, 22, 23, 24, 25]. In [19], Abualrub et al. investigated cyclic DNA
sequences over a finite field $\mathbb{F}_{4}$. They studied DNA codes with
reversible-complement constraint over $\mathbb{F}_{4}$. In [20], Oztas and
Siap studied reversible DNA codes over the finite field $\mathbb{F}_{16}$ with
$16$ elements by introducing a new family of lifted polynomials. Later, in
[21], Srinivasulu and Bhaintwal investigated DNA codes with reversible-
complement constraint over a finite chain ring
$\mathbb{F}_{4}+u\mathbb{F}_{4}$ with $16$ elements and $u^{2}=0$. In [22],
Bayram et al. studied DNA codes reversible constraint over a finite non-chain
ring $\mathbb{F}_{4}+v\mathbb{F}_{4}$ with $v^{2}=v$. In [23], Dinh et al.
proposed a construction of DNA codes with reversible constraint over a finite
non-chain ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4}$ with $16$ elements and
$u^{2}=1$. In [24], Liu studied reversible DNA codes over a finite chain ring
$\mathbb{F}_{4}[u]/\langle u^{3}\rangle$ with $64$ elements and $u^{3}=0$.
They also investigated a construction method of reversible and reversible-
complement DNA codes over a finite non-chain ring
$\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ with $16$ elements and $v^{2}=v$ in [25]. We
aim to investigate DNA codes over
$\mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4}$ with the maximum number
of constraints and preferably better code-rate compared to existing work in
the literature.

In this paper, we study DNA sequences over a finite ring
$R=\mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4}$ with $64$ elements and
$u^{3}=1$. We propose a design strategy for DNA codes with reversible,
reversible-complement, $\frac{2}{3}$-GC-content, and homopolymer 2-run-length
constraints over $R$. To be more specific, the main contributions of this
paper are as follows:

  * •

In this paper, we investigate a distance conserving bijective map $\psi$
between the ring $R$ and $\\{\text{A, C, G, T}\\}^{3}$. This mapping allows us
to design DNA codes over $R$ with multiple constraints.

  * •

Next, we study different properties of linear codes over $R$ with $u^{3}=1$ to
construct DNA codes with reversible, reversible-complement, homopolymers
$2$-run-length, and $\frac{2}{3}$-GC-content constraints.

  * •

Finally, we propose a novel construction of new DNA codes satisfying multiple
constraints of length $3\cdot 2^{m}$ over the finite non-chain ring $R$ by
using Reed-Muller type generator matrices.

The rest of the paper is organized as follows. In Section II, preliminaries
and notations are presented. In Section III, the distance conserving map, Gau
distance, and their properties are investigated. Then, DNA codes with multiple
constraints are constructed in Section IV. Finally, a comparative analysis is
drawn in Section V.

##  II Preliminaries

In this section, we study the structure of the ring
$R=\mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4}$ with $u^{3}=1$. Then,
we discuss some basic definitions and notations.

Let
$R=\mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4}=\\{a+bu+cu^{2}:a,b,c\in\mathbb{Z}_{4}\mbox{
and }u^{3}=1\\}$. The ring $R$ is a commutative ring with identity and $64$
elements. An arbitrary element $x$ of the ring $R$ can be uniquely expressed
by $x=a+bu+cu^{2}$, where $a,b,c\in\mathbb{Z}_{4}$. If we consider $u^{3}=1$,
ideals of $R$ are not comparable. Thus, the ring $R$ is a non-chain ring. For
the ring $R$, there are $24$ unit elements and $40$ zero divisors. The set of
all $24$ unit elements is $\mathcal{U}$ =
$\\{2u+3u^{2},3u,3u+2u^{2},1,1+2u^{2},1+2u,1+2u+2u^{2},2+u^{2},2+3u^{2},2+u,2+u+2u^{2},2+2u+u^{2},2+2u+3u^{2},2+3u,2+3u+2u^{2},3,3+2u^{2},3+2u,3+2u+2u^{2},u^{2},3u^{2},u,u+2u^{2},2u+u^{2}\\}$.
Now, we define a sequence a = $(a_{1},a_{2},\ldots,a_{n})$ of length $n$ over
$R$. Further, for the sequence a, we define the sequence $\textbf{a}^{r}$ =
$(a_{n},a_{n-1},\ldots,a_{1})$. Now, for any arbitrary matrix $G$ with $k$
rows $\textbf{g}_{i}$ ($i=1,2,\cdots,k$), $\langle G\rangle$ =
$\left\\{\sum_{i=1}^{k}a_{i}\cdot\textbf{g}_{i}:a_{i}\in R\right\\}$ is the
row sub-module of the matrix $G$ over $R$. Again, for any given element $x\in
R$, $\langle x\rangle$ = $\\{a\cdot x:a\in R\\}$. For any positive integer
$n$, a set $C\subseteq R^{n}$ of size $M$ is called a code with parameter
$(n,M,d_{\text{G}})$, where the minimum Gau distance $d_{\text{G}}$ will be
defined in Section III-A.

Figure 1: Rearrangement of the elements of $R$ and the DNA sequences of length
three in a three dimensional box. We denote this three dimensional box by
$\Xi$. Here, an element $a+bu+cu^{2}\in R$ is simply denoted by ‘$abc$’.

A sequence c = $(c_{1},c_{2},\ldots,c_{n})$ of length $n$ over $\Sigma_{DNA}$
= $\\{\text{A, C, T, G}\\}$ is said to be a DNA sequence and also it can be
represented by $c_{1}c_{2}\ldots c_{n}$. For any DNA sequence c =
$c_{1}c_{2}\ldots c_{n}$, the reverse DNA sequence is $\textbf{c}^{r}$ =
$c_{n}c_{n-1}\ldots c_{1}$, and the reverse-complement DNA sequence is
$\textbf{c}^{rc}$ = $c^{c}_{n}c^{c}_{n-1}\ldots c^{c}_{1}$, where
$\text{C}^{c}=$ G, $\text{G}^{c}=$ C, $\text{A}^{c}=$ T and $\text{T}^{c}=$ A.
Also, the GC-content, $w_{GC}(\textbf{c})$, of the DNA sequence c is the total
number of G’s and C’s in the DNA sequence. In a DNA sequence, a homopolymer of
run-length $\ell$ is the maximum number of consecutive positions $\ell$ where
all the positions are $x$ for some $x\in\Sigma_{DNA}$. For example, the DNA
sequence CAAAGGT of length $7$ has a homopolymer of maximum run-length $3$. A
set $C_{DNA}\subseteq\Sigma_{DNA}^{n}$ of size $M$ is called a DNA code with
parameter $(n,M,d_{H})$, where the minimum Hamming distance $d_{H}$ =
$\min\\{d_{H}(\textbf{x},\textbf{y}):\textbf{x}\neq\textbf{y}\mbox{ and
}\textbf{x},\textbf{y}\in C_{DNA}\\}$ and $d_{H}(\textbf{x},\textbf{y})$ is
the Hamming distance between the DNA sequences x and y. The problem of
designing DNA codes is to construct the desired DNA codes satisfying different
constraints. There are four different constraints on $(n,M,d_{H})$ DNA codes
$C$ that are considered in this work:

  * •

The Homopolymer $k$-Run-Length Constraint: In each codeword $\textbf{c}\in C$,
if the maximum possible run-length is $k\ (\geq 1)$, then the DNA code
satisfies the Homopolymer $k$-run-length constraint.

  * •

The $\frac{w}{n}$-GC-Content Constraint: For any DNA code with length $n$, if
all DNA codewords have the same GC-content $w$, then the DNA code satisfies
the $\frac{w}{n}$-GC-content constraint.

  * •

The Reversible Constraint: For any two codewords
$\textbf{c}_{1},\textbf{c}_{2}\in C$ such that
$\textbf{c}_{1}^{r}\neq\textbf{c}_{2}$, the DNA code holds reversible
constraint if $d_{H}(\textbf{c}_{1}^{r},\textbf{c}_{2})\geq d_{H}$.

  * •

The Reversible-Complement Constraint: For all
$\textbf{c}_{1},\textbf{c}_{2}\in C$ such that
$\textbf{c}_{1}^{rc}\neq\textbf{c}_{2}$, the DNA code holds reversible-
complement constraint if $d_{H}(\textbf{c}_{1}^{rc},\textbf{c}_{2})\geq
d_{H}$.

##  III DNA Codes over the Ring $R$

In this section, we study and analyze a distance conserving map
$\psi:R\rightarrow\Sigma_{\text{DNA}}^{3}$ (given in Table I) with some key
properties.

###  III-A Distance Conserving Map

To construct the DNA codes over $R$, we first investigate a one-to-one
correspondence between the length-3 DNA sequences and elements of $R$. We
first rearrange all the $64$ elements of $R$ in a three-dimensional (3-D) box
as given in Fig. 1 to define the distance conserving map $\psi$. In Fig. 1, an
element $a+bu+cu^{2}\in R$ is denoted by ‘$abc$’. For example, $1+3u+2u^{2}$
is denoted by ‘$132$’. In the 3-D box $\Xi$, the elements of $R$ and the DNA
sequences of length three are placed such that the distance $d_{H}$ between
two distinct DNA sequences in a single axis is 1, in two different axes is 2,
and in three different axes is 3. This representation allows us to define the
distance (i.e., Gau distance) between the elements of the ring $R$ so that the
Hamming distance property is preserved.

We consider that the positions of two elements $x$ and $y$ from $R$ are
$(i,j,k)$ and $(i^{\prime},j^{\prime},k^{\prime})$ in the 3-D box $\Xi$ with
$0\leq i,j,k\leq 3$ and $0\leq i^{\prime},j^{\prime},k^{\prime}\leq 3$. Then,
the Gau distance $d_{\text{G}}:R\times R\rightarrow\mathbb{R}$ is defined by
$d_{\text{G}}(x,y)$ = $\text{min}\\{1,(i+3i^{\prime})\mod
4\\}+\text{min}\\{1,(j+3j^{\prime})\mod
4\\}+\text{min}\\{1,(k+3k^{\prime})\mod 4\\}$. For any integers $t$ and
$t^{\prime}$ ($0\leq t,t^{\prime}\leq 3$), we can see that
$\min\\{1,(t+3t^{\prime})\mod 4\\}\geq 0$, $\min\\{1,(t+3t^{\prime})\mod 4\\}$
= $0$ iff $t=t^{\prime}$, and $\min\\{1,(t+3t^{\prime})\mod 4\\}$ =
$\min\\{1,(t^{\prime}+3t)\mod 4\\}$. Also, for any integers $t$, $t^{\prime}$
and $t^{\prime\prime}$ ($0\leq t,t^{\prime},t^{\prime\prime}\leq 3$), we can
observe that $\min\\{1,(t+3t^{\prime\prime})\mod
4\\}\leq\min\\{1,(t+3t^{\prime})\mod
4\\}+\min\\{1,(t^{\prime}+3t^{\prime\prime})\mod 4\\}$. Therefore, the mapping
$d_{\text{G}}$ satisfies the identity of indiscernible, symmetry, and triangle
inequality. In other words, the mapping $d_{\text{G}}$ is a metric defined
over $R$.

For any two vectors $\textbf{x}=(x_{1},x_{2},\cdots,x_{n})$ and
$\textbf{y}=(y_{1},y_{2},\cdots,y_{n})$ of equal length $n$ over $R$, the Gau
distance between x and y is defined by
$d_{\text{G}}(\textbf{x},\textbf{y})=\sum_{i=1}^{n}d_{\text{G}}(x_{i},y_{i})$.
For any code $C$ over $R$, the minimum Gau distance $d_{\text{G}}$ is defined
by
$d_{\text{G}}=\text{min}\\{d_{\text{G}}(\textbf{x},\textbf{y}):\textbf{x},\textbf{y}\in
C\ \text{with}\ \textbf{x}\neq\textbf{y}\\}$. For example, we consider
$\textbf{x}=(2u,1+3u^{2},2+u^{2},3)\in R^{4}$ and
$\textbf{y}=(1+2u+u^{2},3u,2+2u,3u+2u^{2})\in R^{4}$. From the 3-D box $\Xi$,
we have $d_{\text{G}}(2u,1+2u+u^{2})=2$, $d_{\text{G}}(1+3u^{2},3u)=3$,
$d_{\text{G}}(2+u^{2},2+2u)=2$ and $d_{\text{G}}(3,3u+2u^{2})=3$. Thus, the
Gau distance between x and y is $d_{\text{G}}(\textbf{x},\textbf{y})$ =
$d_{\text{G}}(2u,1+2u+u^{2})+d_{\text{G}}(1+3u^{2},3u)+d_{\text{G}}(2+u^{2},2+2u)$
$+d_{\text{G}}(3,3u+2u^{2})$ = $2+3+2+3$ = $10$.

We now define the distance conserving map
$\psi:R\rightarrow\Sigma_{\text{DNA}}^{3}$ such that, for any $x\in R$,

$\displaystyle\psi(x)^{c}=\psi(x+(2+2u+2u^{2})),\mbox{ and }$ (1)
$\displaystyle\psi(x)^{r}=\psi(x^{r}),$ (2)

where $x^{r}=c+bu+au^{2}$ for the element $x=a+bu+cu^{2}\in R$. By using the
symmetry between the ring element and the DNA sequence of length $3$, we have
developed the distance conserving mapping $\psi$ defined over the ring $R$
with $u^{3}=1$. We illustrate this bijective mapping
$\psi:R\rightarrow\Sigma_{\text{DNA}}^{3}$ in Table I. Note that this
bijective mapping $\psi$ is not unique. For any $x\in R$, it is clear that
$\psi^{-1}\left(\psi(x)\right)$, $\psi^{-1}\left(\psi(x)^{c}\right)$,
$\psi^{-1}\left(\psi(x)^{r}\right)$, and $\psi^{-1}\left(\psi(x)^{rc}\right)$
are in the ring $R$. For any $\textbf{x}=(x_{1},x_{2},\cdots,x_{n})$ over $R$,
we will define following:

$\begin{array}[]{l}\psi^{-1}\left(\psi(\textbf{x})\right)=\left(\psi^{-1}\left(\psi(x_{1})\right),\psi^{-1}\left(\psi(x_{2})\right),\cdots\right.\\\
\hskip 156.49014pt\left.\cdots,\psi^{-1}\left(\psi(x_{n})\right)\right),\\\
\psi^{-1}\left(\psi(\textbf{x})^{r}\right)=\left(\psi^{-1}\left(\psi(x_{n})^{r}\right),\psi^{-1}\left(\psi(x_{n-1})^{r}\right),\cdots\right.\\\
\hskip
156.49014pt\left.\cdots,\psi^{-1}\left(\psi(x_{1})^{r}\right)\right),\\\
\psi^{-1}\left(\psi(\textbf{x})^{c}\right)=\left(\psi^{-1}\left(\psi(x_{1})^{c}\right),\psi^{-1}\left(\psi(x_{2})^{c}\right),\cdots\right.\\\
\hskip
142.26378pt\left.\cdots,\psi^{-1}\left(\psi(x_{n})^{c}\right)\right)\mbox{ and
}\\\
\psi^{-1}\left(\psi(\textbf{x})^{rc}\right)=\left(\psi^{-1}\left(\psi(x_{n})^{rc}\right),\psi^{-1}\left(\psi(x_{n-1})^{rc}\right),\cdots\right.\\\
\hskip
150.79968pt\left.\cdots,\psi^{-1}\left(\psi(x_{1})^{rc}\right)\right).\end{array}$

For any code $C$ of length $n$ over $R$, the DNA code of length $3n$ is given
by $\psi(C)=\\{\psi(\textbf{x}):\mbox{ each }\textbf{x}\in
C\\}\subseteq\Sigma_{\text{DNA}}^{3n}$.

TABLE I: Bijective map $\psi:R\rightarrow\Sigma_{DNA}^{3}$ for all $x\in R$. $x$ | $\psi(x)$ | $x$ | $\psi(x)$ | $x$ | $\psi(x)$ | $x$ | $\psi(x)$ | $x$ | $\psi(x)$ | $x$ | $\psi(x)$
---|---|---|---|---|---|---|---|---|---|---|---
$0$ | GAG | $3u+2u^{2}$ | AGT | $1+2u+u^{2}$ | TTT | $2+u$ | TCA | $2+3u^{2}$ | TAG | $3+3u+u^{2}$ | GGC
$u$ | ACA | $3u^{2}$ | AAG | $1+3u+u^{2}$ | CGC | $2+2u$ | TCG | $2+u+3u^{2}$ | TTA | $3+2u^{2}$ | GAT
$2u$ | GTG | $u+3u^{2}$ | ACG | $1+2u^{2}$ | CAT | $2+3u$ | TGA | $2+2u+3u^{2}$ | TTG | $3+u+2u^{2}$ | ATT
$3u$ | AGA | $2u+3u^{2}$ | ATG | $1+u+2u^{2}$ | CCT | $2+u^{2}$ | TAC | $2+3u+3u^{2}$ | TGG | $3+2u+2u^{2}$ | GTT
$u^{2}$ | AAC | $3u+3u^{2}$ | AGG | $1+2u+2u^{2}$ | CTT | $2+u+u^{2}$ | TCC | $3$ | GAA | $3+3u+2u^{2}$ | GGT
$u+u^{2}$ | ACC | $1$ | CAA | $1+3u+2u^{2}$ | CGT | $2+2u+u^{2}$ | TTC | $3+u$ | GCA | $3+3u^{2}$ | AAA
$2u+u^{2}$ | ATC | $1+u$ | CCA | $1+3u^{2}$ | CAG | $2+3u+u^{2}$ | TGC | $3+2u$ | GTA | $3+u+3u^{2}$ | GCG
$3u+u^{2}$ | AAT | $1+2u$ | CTA | $1+u+3u^{2}$ | CCG | $2+2u^{2}$ | CAC | $3+3u$ | GGA | $3+2u+3u^{2}$ | ATA
$2u^{2}$ | AGC | $1+3u$ | TAA | $1+2u+3u^{2}$ | CTG | $2+u+2u^{2}$ | TCT | $3+u^{2}$ | GAC | $3+3u+3u^{2}$ | GGG
$u+2u^{2}$ | ACT | $1+u^{2}$ | TAT | $1+3u+3u^{2}$ | CGG | $2+2u+2u^{2}$ | CTC | $3+u+u^{2}$ | GCC |  |
$2u+2u^{2}$ | GCT | $1+u+u^{2}$ | CCC | $2$ | CGA | $2+3u+2u^{2}$ | TGT | $3+2u+u^{2}$ | GTC |  |

###  III-B Properties of the Distance Conserving Map

In this section, we investigate the properties of distance conserving map and
Gau distance.

######  Property 1 (Distance Conserving Property)

The map $\psi$ is a distance conserving map from $(R^{n},d_{\text{G}})$ to
$\left(\Sigma_{\text{DNA}}^{3n},d_{H}\right)$.

######  Property 2 (Linearity Property w.r.t. Reverse)

For two length-$n$ vectors x and y in $R^{n}$, the map $\psi$ satisfies
$\psi^{-1}\left(\psi(a\textbf{x}+b\textbf{y})^{r}\right)=a\psi^{-1}\left(\psi(\textbf{x})^{r}\right)+b\psi^{-1}\left(\psi(\textbf{y})^{r}\right)$.

From equation (1), we have the following result.

######  Theorem 1

For any given generator matrix $\mathcal{G}$ over $R$, the code
$\psi(\left\langle\mathcal{G}\right\rangle)$ is closed under complement DNA
sequences if ${\bf 2+2u+2u^{2}}\in\left\langle\mathcal{G}\right\rangle$, where
${\bf 2+2u+2u^{2}}$ denotes the vector of length $n$ with all $2+2u+2u^{2}$
element.

From Property 2, we obtain Theorem 2. Later, we will apply Theorem 2 to design
DNA codes with reversible and reversible-complement constraints over $R$.

######  Theorem 2

The DNA code $\psi(\left\langle\mathcal{G}\right\rangle)$ is closed under
reverse DNA sequence if
$\textbf{g}^{r}=\psi^{-1}(\psi(\textbf{g})^{r})\in\left\langle\mathcal{G}\right\rangle$
for each row g.

###### Proof:

For a matrix $\mathcal{G}$ with $k$ rows $\textbf{g}_{1}$, $\textbf{g}_{2}$,
$\ldots$, $\textbf{g}_{k}$, let
$\textbf{y}\in\left\langle\mathcal{G}\right\rangle$. To prove this result, we
only need to show
$\psi(\textbf{y})^{r}\in\psi(\left\langle\mathcal{G}\right\rangle)$. Since
$\textbf{y}\in\left\langle\mathcal{G}\right\rangle$, there exist
$a_{1},a_{2},\cdots,a_{k}\in R$ such that
$\textbf{y}=\sum_{i=1}^{k}a_{i}\cdot\textbf{g}_{i}$. By using the property of
the operation $r$ on the element of $R$, we have
$\textbf{y}^{r}=\sum_{i=1}^{k}a_{i}\cdot\textbf{g}_{i}^{r}$. According to the
given condition, each
$\textbf{g}_{i}^{r}\in\left\langle\mathcal{G}\right\rangle$ for
$i=1,2,\cdots,k$. Thus,
$\textbf{y}^{r}=\sum_{i=1}^{k}a_{i}\cdot\textbf{g}_{i}^{r}\in\left\langle\mathcal{G}\right\rangle$.
By applying Property 2, we have
$\textbf{y}^{r}\in\left\langle\mathcal{G}\right\rangle\Rightarrow\psi^{-1}\left(\psi(\textbf{y})^{r}\right)\in\left\langle\mathcal{G}\right\rangle\Rightarrow\psi(\textbf{y})^{r}\in\psi(\left\langle\mathcal{G}\right\rangle)$.
∎

From Theorem 1 and Theorem 2, we have Corollary 1.

######  Corollary 1

For a given $\mathcal{G}$ over $R$, the code
$\psi(\left\langle\mathcal{G}\right\rangle)$ satisfies the reversible-
complement constraint if $\psi(\left\langle\mathcal{G}\right\rangle)$ is
closed under both reverse and reverse-complement DNA sequences.

Consequently, we have the following result.

######  Theorem 3

Let $C$ be a linear $(n,M,d_{\text{G}})$ code over $R$ and $\mathcal{G}$ be
the associated generator matrix of $C$ such that the rows of $\mathcal{G}$
hold the conditions described in Theorem 1 and Theorem 2, where $n,M$, and
$d_{\text{G}}$ denote the length, size, and the minimum Gau distance of the
code $C$. Then, $\psi(C)$ is a $(3n,M,d_{H})$ DNA code with length $3n$, the
size $M$, and the minimum Hamming distance $d_{H}$ = $d_{\text{G}}$. The DNA
code $\psi(C)$ holds reversible and reversible-complement constraints.

###### Proof:

From Property 1, Theorem 1 and Theorem 2, the proof follows. ∎

For any $z$ from $\\{2,2u,2u^{2}\\}$, we have the ideal generated by $z$ given
by $\left\langle
z\right\rangle=\\{0,2,2u,2u^{2},2+2u,2u+2u^{2},2+2u^{2},2+2u+2u^{2}\\}$. We
have discussed the following cases.
Case-I: $y\in R\backslash\langle z\rangle$: In this case, the GC-content of
$\psi(x)$ is not identical for all $x\in\langle y\rangle$. Also, all
length-$3n$ DNA sequences defined over $\psi(\langle y\rangle)$ do not avoid
the homopolymers.
Case-II: $y\in\\{2+2u,2u+2u^{2},2+2u^{2},2+2u+2u^{2}\\}$: In this case, the
GC-content of $\psi(x)$ is identical for all $x\in\langle y\rangle$, and also,
all length-$3n$ DNA sequences defined over $\psi(\langle y\rangle)$ are free
from the homopolymers of run-length more than 2. In this case, we can observe
the following two sub-cases.
Sub-case-I: $y\in\\{2+2u,2u+2u^{2},2+2u^{2}\\}$: In this sub-case, the element
$2+2u+2u^{2}\notin\langle y\rangle$, and therefore, we cannot construct DNA
codes with reversible-complement constraint from linear code over $\langle
y\rangle$ using Theorem 1.
Sub-case-II: $y=2+2u+2u^{2}$: In this sub-case, $\langle 2+2u+2u^{2}\rangle$ =
$\\{0,2+2u+2u^{2}\\}$. The code-rate of any DNA code defined over $\langle
2+2u+2u^{2}\rangle$ is significantly small.

Thus, we are interested to study the properties of DNA codes defined over
$\psi(\left\langle z\right\rangle)$ for $z\in\\{2,2u,2u^{2}\\}$. Now, we
investigate homopolymer property for the obtained DNA sequences in Lemma 1.

######  Lemma 1

For any $z\in\\{2,2u,2u^{2}\\}$, consider a vector $\textbf{x}\in\left\langle
z\right\rangle^{n}$. The DNA sequence $\psi(\textbf{x})$ is free of all
homopolymers of run-length more than $2$.

###### Proof:

For given any $z\in\\{2,2u,2u^{2}\\}$, the proof follows from the principle of
mathematical induction on $n$. Base Case: For any $x\in\left\langle
z\right\rangle$, $\psi(\langle x\rangle)$ =
$\left\\{GAG,CGA,GTG,AGC,TCG,\right.$ $\left.CAC,GCT,CTC\right\\}$, and the
DNA image $\psi(x)$ is free of all homopolymers of run-length more than $2$.
Inductive Step: For a positive integer $m$, consider
$(x_{1},x_{2},\ldots,x_{m})\in\left\langle z\right\rangle^{m}$. Then, assume
the DNA image $\psi((x_{1},x_{2},\ldots,x_{m}))$ is independent from all
homopolymers of run-length more than $2$. Now, for $x_{m+1}\in\left\langle
z\right\rangle$, the vector $(x_{1},x_{2},\ldots,x_{m},x_{m+1})\in\left\langle
z\right\rangle^{m+1}$. In this case, one can observe the following.

  1. 1.

From the assumption, more than two consecutive positions are not identical in
$\psi((x_{1},x_{2},\ldots,x_{m}))$.

  2. 2.

For any vector $(x_{m},x_{m+1})\in\left\langle z\right\rangle^{2}$, more than
two consecutive positions are not identical in $\psi((x_{m},x_{m+1}))$.

And thus, more than two consecutive positions are not identical in the DNA
image $\psi((x_{1},x_{2},\ldots,x_{m+1}))$. So, from the definition of
homopolymers, the DNA image $\psi((x_{1},x_{2},\ldots,x_{m+1}))$ is free from
all homopolymers of run-length more than $2$. Now, from mathematical induction
on $n$, the result follows. ∎

We now investigate some important properties of the constructed DNA sequences
as follows.

######  Lemma 2

For any $z\in\\{2,2u,2u^{2}\\}$, if $\textbf{x}\in\left\langle
z\right\rangle^{n}$ then the GC-content of the $3n$-length DNA sequence
$\psi(\textbf{x})$ is $2n$.

###### Proof:

For any given $z\in\\{2,2u,2u^{2}\\}$, if $x\in\left\langle z\right\rangle$,
then the GC-content of $\psi(x)$ is $w_{GC}(\psi(x))$ = $2$. Thus, for x =
$(x_{1},x_{2},\ldots,x_{n})\in\left\langle z\right\rangle^{n}$ the GC-content
of the DNA sequence $\psi(\textbf{x})$ is $w_{GC}(\textbf{x})$ =
$\sum_{i=1}^{n}w_{GC}(\psi(x_{i}))$ = $2n$. ∎

######  Theorem 4

For any $z\in\\{2,2u,2u^{2}\\}$, we consider a linear code with the generator
matrix $\mathcal{G}$ over $\left\langle z\right\rangle$ such that the rows of
$\mathcal{G}$ hold the conditions described in Theorem 1 and Theorem 2. Then,
we have the following results:

  1. 1.

The DNA code $\psi(\left\langle\mathcal{G}\right\rangle)$ satisfies
reversible, and reversible-complement constraints.

  2. 2.

The GC-content of each DNA codeword of length $3n$ is $2n$.

  3. 3.

All DNA codewords are free from all homopolymers of run-length more than $2$.

###### Proof:

From Lemma 1, all the DNA sequences in the DNA code
$\psi(\left\langle\mathcal{G}\right\rangle)$ are independent from homopolymers
of run-length more than $2$. Further, from Lemma 2, all the DNA sequences in
the DNA code $\psi(\left\langle\mathcal{G}\right\rangle)$ satisfy
$\frac{2}{3}$-GC-content constraint. The result on reversible and reversible-
complement properties follow from Theorem 1 and Theorem 2. ∎

##  IV Proposed Constructions of DNA Codes

In this section, we present new constructions of DNA codes by using Reed-
Muller type codes over $R$.

For a given positive integer $m\ (\geq 1)$, the generator matrix
$\mathcal{G}_{1,m}$ of the Reed-Muller type code $\mathcal{R}(1,m)$ of length
$2^{m}$ is

$\mathcal{G}_{1,i+1}=\begin{bmatrix}\mathcal{G}_{1,i}&\mathcal{G}_{1,i}\\\
\textit{{0}}_{2^{i}}&\textit{{z}}_{2^{i}}\end{bmatrix};\ 1\leq i\leq m-1,$ (3)

where $\mathcal{G}_{1,1}=\begin{bmatrix}z&z\\\ 0&z\end{bmatrix}$,
$\textit{{0}}_{2^{i}}=\underbrace{[0\ 0\ \cdots\ 0]}_{2^{i}}$,
$\textit{{z}}_{2^{i}}=\underbrace{[z\ z\ \cdots\ z]}_{2^{i}}$ with $z\in R$.

Clearly, the number of rows and number of columns of the generator matrix
$\mathcal{G}_{1,m}$ are $(m+1)$ and $2^{m}$.

###### Remark 1

For any positive integer $m$ and $z\in R$, $\textbf{g}_{i}$ is the $i$-th row
of $\mathcal{G}_{1,m}$ then $\textbf{g}_{i}^{r}$ = $\textbf{g}_{1}$ and
$\textbf{g}_{i}^{r}$ = $\textbf{g}_{1}-\textbf{g}_{i}$ for $i=2,3,\ldots,m+1$.
Thus, $\textbf{g}_{i}^{r}\in\langle\mathcal{G}_{1,m}\rangle$ for
$i=1,2,\ldots,m+1$.

###### Remark 2

For any positive integer $m$ and
$z\in\mathcal{Z}\backslash\\{2u+2u^{2},2+2u^{2},2+2u,1+u+2u^{2},1+2u+u^{2},2+u+u^{2},2+3u+3u^{2},3+2u+3u^{2},3+3u+2u^{2},u+3u^{2},3u+u^{2},1+3u^{2},1+3u,3+u^{2},3+u\\}$,
we have $2+2u+2u^{2}\in\langle z\rangle$. Thus, ${\bf
2+2u+2u^{2}}\in\langle\mathcal{G}_{1,m}\rangle$ as the vector with all $z$ is
the row of $\mathcal{G}_{1,m}$, where $\mathcal{Z}$ is the collection of all
zero divisors of $R$.

Consequently, based on Theorem 3, we obtain Theorem 5, Theorem 6, Theorem 7
and Theorem 8.

######  Theorem 5

For $z\in\mathcal{U}$, we consider the code $\mathcal{R}(1,m)$ over $R$. Then,
the length, size and minimum Gau distance are $2^{m}$, $64^{m+1}$ and
$d_{\text{G}}=2^{m-1}$.

###### Proof:

For the $m$-th iteration, by using the principle of mathematical induction on
$m$, the code length of $\mathcal{R}(1,m)$ is $2^{m}$. For $z\in\mathcal{U}$,
if a matrix $\mathcal{G}_{1,i}$ over $R$ can be reduced to $[\textbf{I}_{k}\
\textbf{G}_{1,2}]$ then the size of $\langle\mathcal{G}_{1,i}\rangle$ is
$64^{k}$, where $\textbf{G}_{1,2}$ is a matrix over $R$. For any $x,y\in R$,
the distance $d_{\text{G}}(x,y)\geq d_{H}(x,y)$. Note that $d_{H}(x,y)$ refers
to the Hamming distance between the ring elements $x$ and $y$ (not to be
confused with the Hamming distance between DNA sequences). Thus, for the code
$\mathcal{R}(1,m)$ with the minimum Gau distance $d_{\text{G}}$ and the
minimum Hamming distance $2^{m-1}$, $d_{\text{G}}\geq 2^{m-1}$. But,
$\textit{{0}}_{2^{m}}$ and $\textit{{0}}_{2^{m-1}}||\textit{{3}}_{2^{m-1}}$
are the codewords of $\mathcal{R}(1,m)$ for any positive integer $m$, where
$||$ denotes the concatenation of two sequences. Then,
$d_{\text{G}}(\textit{{0}}_{2^{m}},\textit{{0}}_{2^{m-1}}||\textit{{3}}_{2^{m-1}})$
= $d_{\text{G}}(\textit{{0}}_{2^{m-1}},\textit{{3}}_{2^{m-1}})$ = $2^{m-1}$.
Thus, the minimum Gau distance is $d_{\text{G}}=2^{m-1}$. ∎

######  Theorem 6

For the code $\mathcal{R}(1,m)$ over $R$, there exists a DNA code
$\psi(\mathcal{R}(1,m))$, that satisfies both reversible and reversible-
complement constraints with the length $3\cdot 2^{m}$, size $64^{m+1}$ and
minimum Hamming distance $2^{m-1}$.

###### Proof:

The sequence length, size and minimum Hamming distance follow from Theorem 3
and Theorem 5. The reversible and reversible-complement constraints follow
from Remark 1, Remark 2, Theorem 1 and Theorem 2. ∎

###### Example 1

For $m=2$ and $z=3$, the $(n=12,M=262144,d_{H}=2)$ DNA code
$\psi(\mathcal{R}(1,2))$ holds the reversible and reversible-complement
constraints with the relative minimum Hamming distance $0.167$ and the code-
rate $0.750$.

######  Theorem 7

For $z\in\\{2,2u,2u^{2}\\}$, the Reed-Muller type code $\mathcal{R}(1,m)$ over
$R$, the length, size and minimum Gau distance are $2^{m}$, $8^{m+1}$ and
$d_{\text{G}}=2^{m-1}$.

###### Proof:

The proof is similar to the proof of Theorem 5. Also, $\mathcal{G}_{1,m}$ can
be reduced into $[2\textbf{I}_{k}\ 2\textbf{G}_{1,2}]$, since there exists
$y\in R$ such that $x=2y$ for each $x\in\langle z\rangle$. Then, the size of
$\langle\mathcal{G}_{1,m}\rangle$ is $8^{m+1}$, where $\textbf{G}_{1,2}$ is
the matrix over $R$. ∎

######  Theorem 8

For $z\in\\{2,2u,2u^{2}\\}$ and the Reed-Muller type code $\mathcal{R}(1,m)$
over $R$, the DNA code $\psi(\mathcal{R}(1,m))$, satisfies Homopolymer
$2$-run-length, $\frac{2}{3}$-GC-content, reversible, and reversible-
complement constraints with length $3\cdot 2^{m}$, size $8^{m+1}$ and minimum
Hamming distance $2^{m-1}$.

###### Proof:

The proof follows from Theorem 4. ∎

###### Example 2

For $m=2$ and $z=2u$, the $(n=12,M=512,d_{H}=2)$ DNA code
$\psi(\mathcal{R}(1,2))$ satisfies the Homopolymer $2$-run-length,
$\frac{2}{3}$-GC-content, reversible and reversible-complement constraints
with the relative minimum Hamming distance of $C$ is $0.167$ and the code-rate
is $0.375$.

###### Remark 3

Note that the code-rate for above Reed-Muller type codes approaches zero as
length approaches infinity. Thus, the code-rate of the constructed DNA codes
vanishes for the large length. However, if we consider a linear code such that
the code-rate does not vanish for the large length (e.g., Hamming type codes),
then the code-rate for the generated DNA codes will also not vanish for the
large length. Therefore, we can obtain DNA codes satisfying all four
constraints such that the code-rate does not vanish for the large length.

Our proposed design method is valid for DNA codes of length $3n$ (given in
Theorem 3), where $n$ is the length of the code defined over $R$. In
particular, when we consider the Reed-Muller type code of length $2^{m}$, we
can construct DNA codes of length $3\cdot 2^{m}$ given in Theorem 6.

TABLE II: Comparison with Existing Works Ring/Field and
DNA code |  Parameter
$(n,M,d_{H})$ |  Code
Rate | R | RC | RLF
---|---|---|---|---|---
$\mathbb{Z}_{4}[u]/\langle u^{3}\mbox{-}1\rangle$; Example 1 |  $(12,2^{18},2)$ |  $0.750$ | Yes | Yes | -
$\mathbb{Z}_{4}[u]/\langle u^{3}\mbox{-}1\rangle$; Example 2 |  $(12,512,2)$ |  $0.375$ | Yes | Yes | Yes
$\mathbb{F}_{4}$; [19, Table 1] |  $(7,256,3)$ |  $0.571$ | - | Yes | -
$\mathbb{F}_{16}$; [20, Example 4.6] |  $(18,4096,5)$ |  $0.333$ | - | Yes | -
$\mathbb{Z}_{4}[u]/\langle u^{2}\mbox{-}(2\mbox{+}2u)\rangle$;
[14, Table II] |  $(16,1024,8)$ |  $0.313$ | Yes | Yes | -
$\mathbb{F}_{4}[u]/\langle u^{3}\rangle$;
[24, Example 2(3)] |  $(15,256,10)$ |  $0.267$ | Yes | Yes | -
$\mathbb{F}_{4}[u]/\langle u^{2}\rangle$;
[21, Example 3] |  $(12,64,4)$ |  $0.250$ | - | Yes | -
$\mathbb{F}_{16}$; [20, Example 4.6] |  $(18,256,6)$ |  $0.222$ | Yes | - | -
$\mathbb{F}_{4}[u]/\langle u^{2}\mbox{-}u\rangle$;
[22, Example 5] |  $(20,256,8)$ |  $0.200$ | Yes | - | -
$\mathbb{Z}_{4}[u]/\langle u^{2}\mbox{-}u\rangle$;
[25, Example 2] |  $(30,256,5)$ |  $0.133$ | Yes | Yes | -
Terms R, RC, and RLF refer to the $(n,M,d_{H})$ DNA codes satisfying
reversible, reversible-complement, and Homopolymer $2$-run-length constraints,
respectively.

##  V Discussions and Comparisons

In this paper, for $z\in\mathcal{U}$, we have proposed algebraic construction
for DNA codes $\psi(\mathcal{R}(1,m))$ that satisfies reversible, and
reversible-complement constraints in Theorem 6 with code-rate $(m+1)/2^{m}$.
For $m=2$, an example of $(12,262144,2)$ reversible and reversible-complement
DNA code is illustrated in Example 1 with code-rate $0.750$. Also, for
$z\in\\{2,2u,2u^{2}\\}$, we have proposed algebraic construction for DNA codes
$\psi(\mathcal{R}(1,m))$ that satisfies reversible, reversible-complement,
$\frac{2}{3}$-GC-content and Homopolymer $2$-run-length constraints in Theorem
8 with code-rate $\frac{m+1}{3\cdot 2^{m}}\log_{4}8$. An example of such DNA
code is illustrated in Example 1 with parameter $(12,512,2)$ and code-rate
$0.375$. A comparison among properties of DNA codes is given in Table II.

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