Searcharxiv⌕ Search

arXiv subjects

Aixian Zhang

Publications and source records attributed to Aixian Zhang.

12 recordsLinked to original sources
arXiv2022

Arithmetic autocorrelation distribution of binary $m$-sequences

Binary $m$-sequences are ones with the largest period $n=2^m-1$ among the binary sequences produced by linear shift registers with length $m$. They have a wide range of applications in communication since they have several desirable pseudorandomness such as balance, uniform pattern distribution and ideal (classical) autocorrelation. In his reseach on arithmetic codes, Mandelbaum \cite{9Mand} introduces a 2-adic version of classical autocorrelation of binary sequences, called arithmetic autocorrelation. Later, Goresky and Klapper \cite{3G1,4G2,5G3,6G4} generalize this notion to nonbinary case and develop several properties of arithmetic autocorrelation related to linear shift registers with carry. Recently, Z. Chen et al. \cite{1C1} show an upper bound on arithmetic autocorrelation of binary $m$-sequences and raise a conjecture on absolute value distribution on arithmetic autocorrelation of binary $m$-sequences.

cs.IT↗
arXiv2021

Optimal Combinatorial Neural Codes with Matched Metric $δ_{r}$: Characterization and Constructions

Based on the theoretical neuroscience, G. Cotardo and A. Ravagnavi in \cite{CR} introduced a kind of asymmetric binary codes called combinatorial neural codes (CN codes for short), with a "matched metric" $δ_{r}$ called asymmetric discrepancy, instead of the Hamming distance $d_{H}$ for usual error-correcting codes. They also presented the Hamming, Singleton and Plotkin bounds for CN codes with respect to $δ_{r}$ and asked how to construct the CN codes $\cC$ with large size $|\cC|$ and $δ_{r}(\cC).$ In this paper we firstly show that a binary code $\cC$ reaches one of the above bounds for $δ_{r}(\cC)$ if and only if $\cC$ reaches the corresponding bounds for $d_H$ and $r$ is sufficiently closed to 1. This means that all optimal CN codes come from the usual optimal codes. %(perfect codes, MDS codes or the codes meet the usual Plotkin bound). Secondly we present several constructions of CN codes with nice and flexible parameters $(n,K, δ_r(\cC))$ by using bent functions.

cs.IT↗
arXiv2021

Constructions on Real Approximate Mutually Unbiased Bases

Mutually unbiased bases (MUB) have many applications in quantum information processing and quantum cryptography. Several complex MUB's in $\mathbb{C}^d$ for some dimension $d$ and with larger size have been constructed. On the other hand, real MUB's with larger size are rare which lead to consider constructing approximate MUB (AMUB). In this paper we present a general and useful way to get real AMUB in $\mathbb{R}^{2d}$ from any complex AMUB in $\mathbb{C}^d$. From this method we present many new series of real AMUB's with parameters better than previous results.

quant-ph↗
arXiv2019

On the Constructions of MDS Self-dual Codes via Cyclotomy

MDS self-dual codes over finite fields have attracted a lot of attention in recent years by their theoretical interests in coding theory and applications in cryptography and combinatorics. In this paper we present a series of MDS self-dual codes with new length by using generalized Reed-Solomon codes and extended generalized Reed-Solomon codes as the candidates of MDS codes and taking their evaluation sets as an union of cyclotomic classes. The conditions on such MDS codes being self-dual are expressed in terms of cyclotomic numbers.

cs.IT↗
arXiv2019

An Unified Approach on Constructing of MDS Self-dual Codes via Reed-Solomon Codes

Based on the fundamental results on MDS self-dual codes over finite fields constructed via generalized Reed-Solomon codes \cite{JX} and extended generalized Reed-Solomon codes \cite{Yan}, many series of MDS self-dual codes with different length have been obtained recently by a variety of constructions and individual computations. In this paper, we present an unified approach to get several previous results with concise statements and simplified proofs, and some new constructions on MDS self-dual codes. In the conclusion section we raise two open problems.

cs.IT↗
arXiv2018

A new criterion on k-normal elements over finite fields

The notion of normal elements for finite fields extension has been generalized as k-normal elements by Huczynska et al. [3]. The number of k-normal elements for a fixed finite field extension has been calculated and estimated [3], and several methods to construct k-normal elements have been presented [1,3]. Several criteria on k-normal element have been given [1,2]. In this paper we present a new criterion on k-normal elements by using idempotents and show some examples. Such criterion has been given for usual normal element before [6].

math.NT↗
arXiv2016

Linear Codes over $\mathbb{F}_{q}[x]/(x^2)$ and $GR(p^2,m)$ Reaching the Griesmer Bound

We construct two series of linear codes over finite ring $\mathbb{F}_{q}[x]/(x^2)$ and Galois ring $GR(p^2,m)$ respectively reaching the Griesmer bound. They derive two series of codes over finite field $\mathbb{F}_{q}$ by Gray map. The first series of codes over $\mathbb{F}_{q}$ derived from $\mathbb{F}_{q}[x]/(x^2)$ are linear and also reach the Griesmer bound in some cases. Many of linear codes over finite field we constructed have two Hamming (non-zero) weights.

cs.IT↗
arXiv2016

Linear Codes over Galois Ring $GR(p^2,r)$ Related to Gauss sums

Linear codes over finite rings become one of hot topics in coding theory after Hommons et al.([4], 1994) discovered that several remarkable nonlinear binary codes with some linear-like properties are the images of Gray map of linear codes over $Z_4$. In this paper we consider two series of linear codes $C(G)$ and $\widetilde{C}(G)$ over Galois ring $R=GR(p^2,r)$, where $G$ is a subgroup of $R^{(s)^*}$ and $R^{(s)}=GR(p^2,rs)$. We present a general formula on $N_β(a)$ in terms of Gauss sums on $R^{(s)}$ for each $a\in R$, where $N_β(a)$ is the number of a-component of the codeword $c_β\in C(G) (β\in R^{(s)})$ (Theorem 3.1). We have determined the complete Hamming weight distribution of $C(G)$ and the minimum Hamming distance of $\widetilde{C}(G)$ for some special G (Theorem 3.3 and 3.4). We show a general formula on homogeneous weight of codewords in $C(G)$ and $\widetilde{C}(G)$ (Theorem 4.5) for the special $G$ given in Theorem 3.4. Finally we obtained series of nonlinear codes over $\mathbb{F}_{q} \ (q=p^r)$ with two Hamming distance by using Gray map (Corollary 4.6).

cs.IT↗
arXiv2013

A Family of Five-Weight Cyclic Codes and Their Weight Enumerators

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a family of $p$-ary cyclic codes whose duals have three zeros are proposed. The weight distribution of this family of cyclic codes is determined. It turns out that the proposed cyclic codes have five nonzero weights.

cs.IT↗
arXiv2013

The Weight Enumerator of Three Families of Cyclic Codes

Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems due to their efficient encoding and decoding algorithms. Cyclic codes with many zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a very small number of cases. In general the calculation of the weight distribution of cyclic codes is heavily based on the evaluation of some exponential sums over finite fields. Very recently, Li, Hu, Feng and Ge studied a class of $p$-ary cyclic codes of length $p^{2m}-1$, where $p$ is a prime and $m$ is odd. They determined the weight distribution of this class of cyclic codes by establishing a connection between the involved exponential sums with the spectrum of Hermitian forms graphs. In this paper, this class of $p$-ary cyclic codes is generalized and the weight distribution of the generalized cyclic codes is settled for both even $m$ and odd $m$ alone with the idea of Li, Hu, Feng, and Ge. The weight distributions of two related families of cyclic codes are also determined.

cs.IT↗
arXiv2011

Bounds on and Constructions of Unit Time-Phase Signal Sets

Digital signals are complex-valued functions on $\Z_n$. Signal sets with certain properties are required in various communication systems. Traditional signal sets consider only the time distortion during transmission. Recently, signal sets against both the time and phase distortion have been studied, and are called {\em time-phase} signal sets. Several constructions of time-phase signal sets are available in the literature. There are a number of bounds on time signal sets (also called codebooks). They are automatically bounds on time-phase signal sets, but are bad bounds. The first objective of this paper is to develop better bounds on time-phase signal sets from known bounds on time signal sets. The second objective of this paper is to construct two series of time-phase signal sets, one of which is optimal.

cs.IT↗