arXiv subjects

Bocong Chen

Publications and source records attributed to Bocong Chen.

29 records · Page 2Linked to original sources

arXiv2017

New constructions of MDS codes with complementary duals

Linear complementary-dual (LCD for short) codes are linear codes that intersect with their duals trivially. LCD codes have been used in certain communication systems. It is recently found that LCD codes can be applied in cryptography. This application of LCD codes renewed the interest in the construction of LCD codes having a large minimum distance. MDS codes are optimal in the sense that the minimum distance cannot be improved for given length and code size. Constructing LCD MDS codes is thus of significance in theory and practice. Recently, Jin (\cite{Jin}, IEEE Trans. Inf. Theory, 2016) constructed several classes of LCD MDS codes through generalized Reed-Solomon codes. In this paper, a different approach is proposed to obtain new LCD MDS codes from generalized Reed-Solomon codes. Consequently, new code constructions are provided and certain previously known results in \cite{Jin} are extended.

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arXiv2016

Three new classes of optimal frequency-hopping sequence sets

The study of frequency-hopping sequences (FHSs) has been focused on the establishment of theoretical bounds for the parameters of FHSs as well as on the construction of optimal FHSs with respect to the bounds. Peng and Fan (2004) derived two lower bounds on the maximum nontrivial Hamming correlation of an FHS set, which is an important indicator in measuring the performance of an FHS set employed in practice. In this paper, we obtain two main results. We study the construction of new optimal frequency-hopping sequence sets by using cyclic codes over finite fields. Let $\mathcal{C}$ be a cyclic code of length $n$ over a finite field $\mathbb{F}_q$ such that $\mathcal{C}$ contains the one-dimensional subcode $ \mathcal{C}_0=\{(α,α,\cdots,α)\in \mathbb{F}_q^n\,|\,α\in \mathbb{F}_q\}. $ Two codewords of $\mathcal{C}$ are said to be equivalent if one can be obtained from the other through applying the cyclic shift a certain number of times. We present a necessary and sufficient condition under which the equivalence class of any codeword in $\mathcal{C}\setminus\mathcal{C}_0$ has size $n$. This result addresses an open question raised by Ding {\it et al.} in \cite{Ding09}. As a consequence, three new classes of optimal FHS sets with respect to the Singleton bound are obtained, some of which are also optimal with respect to the Peng-Fan bound at the same time. We also show that the two Peng-Fan bounds are, in fact, identical.

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arXiv2016

Constacyclic symbol-pair codes: lower bounds and optimal constructions

Symbol-pair codes introduced by Cassuto and Blaum (2010) are designed to protect against pair errors in symbol-pair read channels. The higher the minimum pair distance, the more pair errors the code can correct. MDS symbol-pair codes are optimal in the sense that pair distance cannot be improved for given length and code size. The contribution of this paper is twofold. First we present three lower bounds for the minimum pair distance of constacyclic codes, the first two of which generalize the previously known results due to Cassuto and Blaum (2011) and Kai {\it et al.} (2015). The third one exhibits a lower bound for the minimum pair distance of repeated-root cyclic codes. Second we obtain new MDS symbol-pair codes with minimum pair distance seven and eight through repeated-root cyclic codes.

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arXiv2014

A Construction of MDS Quantum Convolutional Codes

In this paper, two new families of MDS quantum convolutional codes are constructed. The first one can be regarded as a generalization of \cite[Theorem 6.5]{GGGlinear}, in the sense that we do not assume that $q\equiv1\pmod{4}$. More specifically, we obtain two classes of MDS quantum convolutional codes with parameters: {\rm (i)}~ $[(q^2+1, q^2-4i+3,1;2,2i+2)]_q$, where $q\geq5$ is an odd prime power and $2\leq i\leq(q-1)/2$; {\rm (ii)}~ $[(\frac{q^2+1}{10},\frac{q^2+1}{10}-4i,1;2,2i+3)]_q$, where $q$ is an odd prime power with the form $q=10m+3$ or $10m+7$ ($m\geq2$), and $2\leq i\leq2m-1$.

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arXiv2014

Polyadic Constacyclic Codes

For any given positive integer $m$, a necessary and sufficient condition for the existence of Type I $m$-adic constacyclic codes is given. Further, for any given integer $s$, a necessary and sufficient condition for $s$ to be a multiplier of a Type I polyadic constacyclic code is given. As an application, some optimal codes from Type I polyadic constacyclic codes, including generalized Reed-Solomon codes and alternant MDS codes, are constructed.

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arXiv2014

Repeated-root constacyclic codes of length $2\ell^mp^n$

For any different odd primes $\ell$ and $p$, structure of constacyclic codes of length $2\ell^mp^n$ over a finite field $\mathbb F_q$ of characteritic $p$ and their duals is established in term of their generator polynomials. Among other results, all linear complimentary dual and self-dual constacyclic codes of length $2\ell^mp^n$ over $\mathbb F_q$ are obtained.

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arXiv2014

Self-dual cyclic codes over finite chain rings

Let $R$ be a finite commutative chain ring with unique maximal ideal $\langle γ\rangle$, and let $n$ be a positive integer coprime with the characteristic of $R/\langle γ\rangle$. In this paper, the algebraic structure of cyclic codes of length $n$ over $R$ is investigated. Some new necessary and sufficient conditions for the existence of nontrivial self-dual cyclic codes are provided. An enumeration formula for the self-dual cyclic codes is also studied.

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arXiv2014

Application of Constacyclic codes to Quantum MDS Codes

Quantum maximal-distance-separable (MDS) codes form an important class of quantum codes. To get $q$-ary quantum MDS codes, it suffices to find linear MDS codes $C$ over $\mathbb{F}_{q^2}$ satisfying $C^{\perp_H}\subseteq C$ by the Hermitian construction and the quantum Singleton bound. If $C^{\perp_{H}}\subseteq C$, we say that $C$ is a dual-containing code. Many new quantum MDS codes with relatively large minimum distance have been produced by constructing dual-containing constacyclic MDS codes (see \cite{Guardia11}, \cite{Kai13}, \cite{Kai14}). These works motivate us to make a careful study on the existence condition for nontrivial dual-containing constacyclic codes. This would help us to avoid unnecessary attempts and provide effective ideas in order to construct dual-containing codes. Several classes of dual-containing MDS constacyclic codes are constructed and their parameters are computed. Consequently, new quantum MDS codes are derived from these parameters. The quantum MDS codes exhibited here have parameters better than the ones available in the literature.

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arXiv2013

Construction of Self-dual Codes over $F_p+vF_p$

In this paper, we determine all self-dual codes over $F_p+vF_p$ ($v^2=v$) in terms of self-dual codes over the finite field $F_p$ and give an explicit construction for self-dual codes over $F_p+vF_p$, where $p$ is a prime.

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arXiv2013

Constacyclic Codes over $F_p+vF_p$

In this paper, we study constacyclic codes over $F_p+vF_p$, where $p$ is an odd prime and $v^2=v$. The polynomial generators of all constacyclic codes over $F_p+vF_p$ are characterized and their dual codes are also determined.

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arXiv2013

Constacyclic Codes over Finite Fields

An equivalence relation called isometry is introduced to classify constacyclic codes over a finite field; the polynomial generators of constacyclic codes of length $\ell^tp^s$ are characterized, where $p$ is the characteristic of the finite field and $\ell$ is a prime different from $p$.

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