arXiv subjects

Shixin Zhu

Publications and source records attributed to Shixin Zhu.

51 records · Page 3Linked to original sources

arXiv2018

A family of constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ and its application to quantum codes

We introduce a Gray map from $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ to $\mathbb{F}_{2}^{2m}$ and study $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}},$ where $u^{2}=0.$ It is proved that the image of a $(1+u)$-constacyclic code length $n$ over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ under the Gray map is a distance-invariant quasi-cyclic code of index $m$ and length $2mn$ over $\mathbb{F}_{2}.$ We also prove that every code of length $2mn$ which is the Gray image of cyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ of length $n$ is permutation equivalent to a binary quasi-cyclic code of index $m.$ Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}.$

cs.IT↗

arXiv2018

Some new bounds on LCD codes over finite fields

In this paper, we show that LCD codes are not equivalent to linear codes over small finite fields. The enumeration of binary optimal LCD codes is obtained. We also get the exact value of LD$(n,2)$ over $\mathbb{F}_3$ and $\mathbb{F}_4$. We study the bound of LCD codes over $\mathbb{F}_q$.

cs.IT↗

arXiv2018

Entanglement-assisted quantum MDS codes constructed from constacyclic codes

Recently, entanglement-assisted quantum error correcting codes (EAQECCs) have been constructed by cyclic codes and negacyclic codes. In this paper, by analyzing the cyclotomic cosets in the defining set of constacyclic codes, we constructed three classes of new EAQECCs which satisfy the entanglement-assisted quantum Singleton bound. Besides, three classes of EAQECCs with maximal entanglement from constacyclic codes are constructed in the meanwhile.

cs.IT↗

arXiv2018

One-Lee weight and two-Lee weight $\mathbb{Z}_2\mathbb{Z}_2[u]$-additive codes

In this paper, we study one-Lee weight and two-Lee weight codes over $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$, where $u^{2}=0$. Some properties of one-Lee weight $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes are given, and a complete classification of one-Lee weight $\mathbb{Z}_2\mathbb{Z}_2[u]$-additive formally self-dual codes is obtained. The structure of two-Lee weight projective $\mathbb{Z}_2\mathbb{Z}_2[u]$ codes is determined. Some optimal binary linear codes are obtained directly from one-Lee weight and two-Lee weight $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes via the extended Gray map.

math.RA↗

arXiv2017

A note on complete classification of $(δ+αu^2)$-constacyclic codes of length $p^k$ over $\F_{p^m}+u\F_{p^m}+u^2\F_{p^m}$

For units $δ$ and $α$ in $\F_{p^m}$, the structure of $(δ+αu^2)$-constacyclic codes of length $p^k$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}$ is studied and self-dual $(δ+αu^2)$-constacyclic codes are analyzed.

cs.IT↗

arXiv2016

The reversible negacyclic codes over finite fields

In this paper, by investigating the factor of the $x^n+1$, we deduce that the structure of the reversible negacyclic code over the finite field $\mathbb{F}_{q}$, where $q$ is an odd prime power. Though studying $q-$cyclotomic cosets modulo $2n$, we obtain the parameters of negacyclic BCH code of length $n=\frac{q^\ell+1}{2}$ , $n=\frac{q^m-1}{2(q-1)}$ and $n=\frac{q^{t\cdot2^τ}-1}{2(q^t+1)}$. Some optimal linear codes from negacyclic codes are given. Finally, we discuss a class of MDS LCD negacyclic codes.

cs.IT↗

arXiv2016

New quantum codes from dual-containing cyclic codes over finite rings

Let $R=\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}+\cdots+u^{k}\mathbb{F}_{2^{m}}$ , where $\mathbb{F}_{2^{m}}$ is a finite field with $2^{m}$ elements, $m$ is a positive integer, $u$ is an indeterminate with $u^{k+1}=0.$ In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over $R$. A new Gray map over $R$ is defined and a sufficient and necessary condition for the existence of dual-containing cyclic codes over $R$ is given. A new family of $2^{m}$-ary quantum codes is obtained via the Gray map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over $R.$ Furthermore, a new family of binary quantum codes is obtained via the Gray map, the trace map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over $R.$

cs.IT↗

arXiv2016

MacWilliams type identities on the Lee and Euclidean weights for linear codes over $\mathbb{Z}_{\ell}$

Motivated by the works of Shiromoto [3] and Shi et al. [4], we study the existence of MacWilliams type identities with respect to Lee and Euclidean weight enumerators for linear codes over $\mathbb{Z}_{\ell}.$ Necessary and sufficient conditions for the existence of MacWilliams type identities with respect to Lee and Euclidean weight enumerators for linear codes over $\mathbb{Z}_{\ell}$ are given. Some examples about such MacWilliams type identities are also presented.

cs.IT↗

arXiv2016

The symbol-pair distance distribution of repeated-root cyclic codes over $\mathbb{F}_{p^m}$

Symbol-pair codes are proposed to protect against pair errors in symbol-pair read channels. One of the most important task in symbol-pair coding theory is to determine the minimum pair-distance of symbol-pair codes. In this paper, we investigate the symbol-pair distances of cyclic codes of length $p^e$ over $\mathbb{F}_{p^m}$. The exact symbol-pair distances of all cyclic codes of such length are determined.

cs.IT↗

arXiv2016

Repeated-root constacyclic codes of length $3lp^{s}$ and their dual codes

Let $p\neq3$ be any prime and $l\neq3$ be any odd prime with $gcd(p,l)=1$. $F_{q}^{*}=\langleξ\rangle$ is decomposed into mutually disjoint union of $gcd(q-1,3lp^{s})$ coset over the subgroup $\langleξ^{3lp^{s}}\rangle$, where $ξ$ is a primitive $(q-1)$th root of unity. We classify all repeated-root constacyclic codes of length $3lp^{s}$ over the finite field $F_{q}$ into some equivalence classes by the decomposition, where $q=p^{m}$, $s$ and $m$ are positive integers. According to the equivalence classes, we explicitly determine the generator polynomials of all repeated-root constacyclic codes of length $3lp^{s}$ over $F_{q}$ and their dual codes. Self-dual cyclic(negacyclic) codes of length $3lp^{s}$ over $F_{q}$ exist only when $p=2$. And we give all self-dual cyclic(negacyclic) codes of length $3l2^{s}$over $F_{2^{m}}$ and its enumeration.

cs.IT↗

arXiv2016

Some results of linear codes over the ring $\mathbb{Z}_4+u\mathbb{Z}_4+v\mathbb{Z}_4+uv\mathbb{Z}_4$

In this paper, we mainly study the theory of linear codes over the ring $R =\mathbb{Z}_4+u\mathbb{Z}_4+v\mathbb{Z}_4+uv\mathbb{Z}_4$. By the Chinese Remainder Theorem, we have $R$ is isomorphic to the direct sum of four rings $\mathbb{Z}_4$. We define a Gray map $Φ$ from $R^{n}$ to $\mathbb{Z}_4^{4n}$, which is a distance preserving map. The Gray image of a cyclic code over $R^{n}$ is a linear code over $\mathbb{Z}_4$. Furthermore, we study the MacWilliams identities of linear codes over $R$ and give the the generator polynomials of cyclic codes over $R$. Finally, we discuss some properties of MDS codes over $R$.

cs.IT↗

arXiv2015

$\mathbb{Z}_p\mathbb{Z}_p[u]$-additive codes

In this paper, we study $\mathbb{Z}_p\mathbb{Z}_p[u]$-additive codes, where $p$ is prime and $u^{2}=0$. In particular, we determine a Gray map from $ \mathbb{Z}_p\mathbb{Z}_p[u]$ to $\mathbb{Z}_p^{ α+2 β}$ and study generator and parity check matrices for these codes. We prove that a Gray map $Φ$ is a distance preserving map from ($\mathbb{Z}_p\mathbb{Z}_p[u]$,Gray distance) to ($\mathbb{Z}_p^{α+2β}$,Hamming distance), it is a weight preserving map as well. Furthermore we study the structure of $\mathbb{Z}_p\mathbb{Z}_p[u]$-additive cyclic codes.

cs.IT↗

arXiv2015

Cyclic DNA codes over F2+uF2+vF2+uvF2

In this work, we study the structure of cyclic DNA codes of arbitrary lengths over the ring R=F2+uF2+vF2+uvF2 and establish relations to codes over R1=F2+uF2 by defining a Gray map between R and R1^2 where R1 is the ring with 4 elements. Cyclic codes of arbitrary lengths over R satisfied the reverse constraint and the reverse-complement constraint are studied in this paper. The GC content constraint is considered in the last.

cs.IT↗

arXiv2014

New quantum MDS codes derived from constacyclic codes

Quantum maximal-distance-separable (MDS) codes form an important class of quantum codes. It is very hard to construct quantum MDS codes with relatively large minimum distance. In this paper, based on classical constacyclic codes, we construct two classes of quantum MDS codes with parameters $$[[λ(q-1),λ(q-1)-2d+2,d]]_q$$ where $2\leq d\leq (q+1)/2+λ-1$, and $q+1=λr$ with $r$ even, and $$[[λ(q-1),λ(q-1)-2d+2,d]]_q$$ where $2\leq d\leq (q+1)/2+λ/2-1$, and $q+1=λr$ with $r$ odd. The quantum MDS codes exhibited here have parameters better than the ones available in the literature.

cs.IT↗

arXiv2014

On the Construction of Optimal Asymmetric Quantum Codes

Constacyclic codes are important classes of linear codes that have been applied to the construction of quantum codes. Six new families of asymmetric quantum codes derived from constacyclic codes are constructed in this paper. Moreover, the constructed asymmetric quantum codes are optimal and different from the codes available in the literature.

cs.IT↗