Linear Codes over the Finite Ring Z15

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1. Introduction

Error correcting codes and error detection codes play an important role in data networks and satellite applications. Most coding theory is interested. Linear code has a clear structure and it is easy to find, understand, edit and decode for codes over finite rings. Since the 1970s, there are many research papers about codes over the finite ring. Several good nonlinear binary codes have been discovered. The circulation code on Z_{4} is composed of a Gary mapping structure [1]. After that, many researchers carried out more and more research on the code of finite ring [2] [3] [4] [5] [6]. The importance of finite rings in algebraic coding theory was established in the early 1990s by observing that some non-linear binary codes actually allow a linear representation of Z_{4} (see [1] [7]). It is also noted that the codes on the ring are particularly useful, if the distance function in the alphabet is not given by the usual Hamming metric, but by the homogeneous weight [8]. Examples of homogeneous weights are Hamming weights on finite fields and Lee weights on Z_{4}. The homogeneous weight can be a natural extension of the Hamming weight of the code over finite rings.

In this paper, we will concern the linear code over the ring Z_{15}, which has
$p\cdot q$ elements and
$p\ne q$. In the first section, we get the ideals of the ring Z_{15} through its non-units and give the Lee weights of elements in Z_{15}. What’s more, we construct a unique expression of an element in Z_{15}. In the second section, we obtain the generate matrix of the dual code of a linear code over the ring
${\mathbb{Z}}_{15}$, give the definition of Gray mapping from
${\mathbb{Z}}_{15}^{n}$ to
${\mathbb{Z}}_{15}^{2n}$ and show that this Gray mapping is distance preserved. In the third section, we prove that the minimal Lee weight of C is equal to the minimal Hamming weight of its Gray Mapping. Further, the linear property of the Gray mapping of a linear code is obtained.

2. The Ring ${\mathbb{Z}}_{15}$

The ring ${\mathbb{Z}}_{15}$ is a non-chain ring, whose units are {1, 2, 4, 7, 8, 11, 13} and non-units are {0, 3, 5, 6, 9, 10, 12}. It has three ideals as follows,

${I}_{\left(0\right)}=\left\{0\right\}$, ${I}_{\left(3\right)}=3{\mathbb{Z}}_{15}=\left\{0,\text{3},\text{6},\text{9},\text{12}\right\}$, ${I}_{\left(5\right)}=5{\mathbb{Z}}_{15}=\left\{0,\text{5},\text{1}0\right\}$

The maximal ideal of the ring ${\mathbb{Z}}_{15}$ are ${I}_{\left(3\right)}$ and ${I}_{\left(5\right)}$, and we have ${\mathbb{Z}}_{15}/{I}_{\left(3\right)}={F}_{3}$ and ${\mathbb{Z}}_{15}/{I}_{\left(5\right)}={F}_{5}$. By the Chinese Remainder Theorem, we have

${\mathbb{Z}}_{15}={I}_{\left(3\right)}\oplus {I}_{\left(5\right)}$.

Thus, for every $a\in {\mathbb{Z}}_{15}$, $a=b\cdot 3+d\cdot 5$. Let $b\in {I}_{\left(3\right)}$ and $d\in {I}_{\left(5\right)}$, then b and d in the expression are unique. The Lee weight of $a\in {\mathbb{Z}}_{15}$ is defined as ${W}_{L}\left(a\right)=2$ if $b\ne 0$ and $d\ne 0$, ${W}_{L}\left(a\right)=0$ if $b=0$ and $d=0$, and ${W}_{L}\left(a\right)=1$ if either $b=0$ or $d=0$. Then the Lee weights of elements in ${\mathbb{Z}}_{15}$ are as follows,

${W}_{L}\left(0\right)=0$ ; ${W}_{L}\left(1\right)=2$ ; ${W}_{L}\left(2\right)=2$ ; ${W}_{L}\left(3\right)=1$ ; ${W}_{L}\left(4\right)=2$ ;

${W}_{L}\left(5\right)=\text{1}$ ; ${W}_{L}\left(6\right)=1$ ; ${W}_{L}\left(7\right)=2$ ; ${W}_{L}\left(8\right)=2$ ; ${W}_{L}\left(9\right)=1$ ;

${W}_{L}\left(1\text{0}\right)=\text{1}$ ; ${W}_{L}\left(11\right)=2$ ; ${W}_{L}\left(12\right)=1$ ; ${W}_{L}\left(13\right)=2$ ; ${W}_{L}\left(14\right)=2$.

3. Linear Codes over the Ring ${\mathbb{Z}}_{15}$

A linear code C over the ring ${\mathbb{Z}}_{15}$ is a additive sub-module. For every code word $x=\left({x}_{1},{x}_{2},\cdots ,{x}_{n}\right)$ and $y=\left({y}_{1},{y}_{2},\cdots ,{y}_{n}\right)$ in C, the inner product of x and y is defined as $\langle x,y\rangle ={x}_{1}{y}_{1}+{x}_{2}{y}_{2}+\cdots +{x}_{n}{y}_{n}$. x is orthogonal to y if $\langle x,y\rangle =0$.

Let C be a linear code over the ring ${\mathbb{Z}}_{15}$ with length n. The dual code of C is ${C}^{{}^{\perp}}=\left\{x\in {\mathbb{Z}}_{15}^{n}|x\cdot y=0,\text{forevery}\text{\hspace{0.17em}}y\in C\right\}$. Thus, ${C}^{{}^{\perp}}$ is also a linear code over the ring ${\mathbb{Z}}_{15}$ with length n. For a codeword $x\in C$, the Lee weight of x is defined as ${W}_{L}\left(x\right)={\displaystyle {\sum}_{i=1}^{n}{W}_{L}\left({x}_{i}\right)}$. For every two code words $x,y\in C$, the Lee distance between x and y is ${d}_{L}\left(x,y\right)={W}_{L}\left(x-y\right)$. The Hamming weight of x is ${W}_{H}\left(x\right)=\left|\left\{{x}_{i}|{x}_{i}\ne 0,1\le i\le n\right\}\right|$, and the Hamming distance between x and y is ${d}_{H}\left(x,y\right)={W}_{H}\left(x-y\right)$.

By Chinese Reminder Theory, the generate matrix of the linear code C over the ring ${\mathbb{Z}}_{15}$ is as follows

$G=\left(\begin{array}{ccccc}{I}_{{k}_{1}}& 5{B}_{1}& 3{A}_{1}& 3{A}_{1}+5{B}_{2}& 3{A}_{3}+5{B}_{3}\\ 0& 3{I}_{{k}_{2}}& 0& 3{A}_{4}& 0\\ 0& 0& 5{I}_{{k}_{3}}& 0& 5{B}_{4}\end{array}\right)$,

in which the elements of ${A}_{i}$ and ${B}_{i}$ belongs to ${\mathbb{Z}}_{15}$. Thus, C is a Abel Group of type ${15}^{{k}_{1}}{5}^{{k}_{2}}{3}^{{k}_{3}}$, and $\left|C\right|={15}^{{k}_{1}}{5}^{{k}_{2}}{3}^{{k}_{3}}$.

Let H be the generate matrix of the dual code ${C}^{{}^{\perp}}$. Then $G{H}^{\text{T}}=0$, in which ${H}^{\text{T}}$ is the rotate matrix of H. Since C is a Abel Group of type ${15}^{{k}_{1}}{5}^{{k}_{2}}{3}^{{k}_{3}}$, H is a Abel Group of type ${15}^{n-\left({k}_{1}-{k}_{2}-{k}_{3}\right)}{5}^{{k}_{3}}{3}^{{k}_{2}}$. Let

$H=\left(\begin{array}{cccc}{M}_{1}& {M}_{2}& {M}_{3}& {I}_{n-\left({k}_{1}+{k}_{2}+{k}_{3}\right)}\\ {Q}_{1}& 0& 3{I}_{{k}_{3}}& 0\\ {Q}_{2}& 5{I}_{{k}_{2}}& 0& 0\end{array}\right)$

By the linear transformation, ${M}_{2}$ and ${M}_{3}$ have can be changed to be the form as

$H=\left(\begin{array}{ccccc}{P}_{1}& 0& 5{R}_{2}& 0& {I}_{n-\left({k}_{1}+{k}_{2}+{k}_{3}+{k}_{4}\right)}\\ {P}_{2}& 3{R}_{1}& 0& {I}_{{k}_{4}}& 0\\ {Q}_{1}& 0& 3{I}_{{k}_{3}}& 0& 0\\ {Q}_{2}& 5{I}_{{k}_{2}}& 0& 0& 0\end{array}\right)$

According to $G{H}^{\text{T}}=0$, we have

$H=\left(\begin{array}{ccccc}12{A}_{3}^{\text{T}}+10{B}_{3}^{\text{T}}& 0& 5{B}_{4}^{\text{T}}& 0& {I}_{n-\left({k}_{1}+{k}_{2}+{k}_{3}+{k}_{4}\right)}\\ 12{A}_{2}^{\text{T}}+10{B}_{2}^{\text{T}}& 3{A}_{4}^{\text{T}}& 0& {I}_{{k}_{4}}& 0\\ 6{A}_{1}^{\text{T}}& 0& 3{I}_{{k}_{3}}& 0& 0\\ 5{B}_{1}^{\text{T}}& 5{I}_{{k}_{2}}& 0& 0& 0\end{array}\right)$

For every $x=\left({x}_{1},{x}_{2},\cdots ,{x}_{n}\right)\in {\mathbb{Z}}_{15}^{n}$, let $x=c\left(x\right)\cdot 3+d\left(x\right)\cdot 5$, in which $c\left(x\right)=\left({c}_{1},{c}_{2},\cdots ,{c}_{n}\right)$, $d\left(x\right)=\left({d}_{1},{d}_{2},\cdots ,{d}_{n}\right)$ and ${x}_{i}={c}_{i}\cdot 3+{d}_{i}\cdot 5$ ( ${c}_{i}\in {I}_{\left(3\right)}$ and ${d}_{i}\in {I}_{\left(5\right)}$ ) for $1\le i\le n$. The Gray mapping from ${\mathbb{Z}}_{15}^{n}$ to ${\mathbb{Z}}_{15}^{2n}$ is

$\Phi \left(c\left(x\right)\cdot 3+d\left(x\right)\cdot 5\right)=\left(c\left(x\right),d(x)\right)$

It’s obvious that the Gray map is a distance preserved mapping from ( ${\mathbb{Z}}_{15}^{n}$, Lee weight) to ( ${\mathbb{Z}}_{15}^{2n}$, Hamming weight).

4. Main Results

Theorem 1. For every $x,y\in {\mathbb{Z}}_{15}^{n}$, we have

${W}_{L}\left(x\right)={W}_{H}\left(\Phi \left(x\right)\right)$, ${d}_{L}\left(x,y\right)={d}_{H}\left(\Phi \left(x\right),\Phi \left(y\right)\right)$.

Therefore, the minimal Lee weight of C is equal to the minimal Hamming weight of $\Phi \left(C\right)$.

Proof: For $x=\left({x}_{1},{x}_{2},\cdots ,{x}_{n}\right)$, let ${x}_{i}={c}_{i}\cdot 3+{d}_{i}\cdot 5\left(1\le i\le n\right)$, in which ${c}_{i}\in {I}_{\left(3\right)}$ and ${d}_{i}\in {I}_{\left(5\right)}$. By the definition of Lee weight and Gray mapping, we have

${W}_{L}\left({x}_{i}\right)={W}_{H}\left(\Phi \left({x}_{i}\right)\right)={W}_{H}\left({c}_{i},{d}_{i}\right)$,

Since $\Phi \left(x\right)=\Phi \left(c\left(x\right)\cdot 3+d\left(x\right)\cdot 5\right)=\left(c\left(x\right),d\left(x\right)\right)$, we have

$\begin{array}{c}{W}_{H}\left(\Phi \left(x\right)\right)={W}_{H}\left(c\left(x\right),d\left(x\right)\right)\\ ={W}_{H}\left(c\left(x\right)\right)+{W}_{H}\left(d\left(x\right)\right)\\ ={\displaystyle {\sum}_{i=1}^{n}{W}_{H}\left({c}_{i}\right)}+{\displaystyle {\sum}_{i=1}^{n}{W}_{H}\left({d}_{i}\right)}\\ ={\displaystyle {\sum}_{i=1}^{n}{W}_{H}\left({c}_{i},{d}_{i}\right)}\\ ={\displaystyle {\sum}_{i=1}^{n}{W}_{L}\left({x}_{i}\right)}={W}_{L}(x)\end{array}$

and

$\begin{array}{c}{d}_{L}\left(x,y\right)={W}_{L}\left(x-y\right)\\ ={W}_{H}\left(\Phi \left(x-y\right)\right)\\ ={W}_{H}\left(\Phi \left(x\right)-\Phi \left(y\right)\right)\\ ={d}_{H}\left(\Phi \left(x\right),\Phi (y)\right)\end{array}$

Thus we get the conclusion. □

Theorem 2. Let C be a linear code over the ring ${\mathbb{Z}}_{15}$ with length n and d is the minimum distance over C. Then $\Phi \left(C\right)$ is a line of code with the Parameter $\left[2n,{k}_{1}+{k}_{2}+{k}_{3},d\right]$.

Proof: For every $x,y\in {\mathbb{Z}}_{15}^{n}$ and ${t}_{1},{t}_{2}\in {\mathbb{Z}}_{15}$, let $x=c\left(x\right)\cdot 3+d\left(x\right)\cdot 5$, $y=c\left(y\right)\cdot 3+d\left(y\right)\cdot 5$. By the definition of gray mapping, we have

$\begin{array}{c}\Phi \left({t}_{1}x+{t}_{2}y\right)=\Phi \left(\left[{t}_{1}c\left(x\right)+{t}_{2}c\left(y\right)\right]\cdot 3+\left[{t}_{1}d\left(x\right)+{t}_{2}d\left(y\right)\right]\cdot 5\right)\\ =\left({t}_{1}c\left(x\right)+{t}_{2}c\left(y\right),{t}_{1}d\left(x\right)+{t}_{2}d\left(y\right)\right)\\ =\left({t}_{1}c\left(x\right),{t}_{1}d\left(x\right)\right)+\left({t}_{2}c\left(y\right),{t}_{2}d\left(y\right)\right)\\ ={t}_{1}\left(c\left(x\right),d\left(x\right)\right)+{t}_{2}\left(c\left(y\right),d\left(y\right)\right)\\ ={t}_{1}\Phi \left(x\right)+{t}_{2}\Phi \left(y\right)\end{array}$.

Thus $\Phi \left(C\right)$ is linear. Since $\left|C\right|={15}^{{k}_{1}}{5}^{{k}_{2}}{3}^{{k}_{3}}$, $\Phi \left(C\right)$ is a linear code with the parameter $\left[2n,{k}_{1}+{k}_{2}+{k}_{3},d\right]$. □

The property of Theorem 2 also applies to the rings which can be decomposed into direct sum of two ideals.

5. Conclusion

By theoretical analysis and derivation, we prove that the minimum Lee weight of a linear code over the ring ${\mathbb{Z}}_{15}$ is equal to the minimum Hamming weight of its Gray mapping. Furthermore, the linear property of its Gray mapping is confirmed. In this paper, we have done some preliminary research work, but there are also many other research contents about linear codes over the ring ${\mathbb{Z}}_{15}$ to be considered, such as weight enumerators, Mac Williams identities and the self-dual codes over the ring ${\mathbb{Z}}_{15}$.

Acknowledgements

This research is supported by Foundation of Langfang Normal University (LSLB201707), Scientific Research Innovation Team of Langfang Normal University (Rings and Algebras with their applications on Error correcting theory), the Key Programs of Scientific Research Foundation of Hebei Educational Committee (Grant No.ZD2019056) and the Key Foundation of Hebei Education Department (ZD2017064).

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