arXiv subjects

Lanqiang Li

Publications and source records attributed to Lanqiang Li.

7 recordsLinked to original sources

arXiv2025

Several new classes of optimal p-ary cyclic codes

Cyclic codes, as a crucial subclass of linear codes, exhibit broad applications in communication systems, data storage systems, and consumer electronics, primarily attributed to their well-structured algebraic properties. Let $p$ denote an odd prime with $p\geq5$, and let $m$ be a positive integer. The primary objective of this paper is to construct three novel classes of optimal $p$-ary cyclic codes, denoted as ${\mathcal{C}_p}(0,s,t)$, which possess the parameters $[{p^m} - 1,{p^m} - 2m - 2,4]$. Here, $s$ is defined as $s = \frac{{{p^m}+1}}{2}$, and $t$ satisfies the condition $2 \le t \le {p^m} - 2$. Notably, one of the constructed classes includes certain known optimal quinary cyclic codes as special cases. Furthermore, for the specific case when $p=5$, this paper additionally presents four new classes of optimal cyclic codes ${\mathcal{C}_5}(0,s,t)$.

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arXiv2024

New QEC codes and EAQEC codes from repeated-root cyclic codes of length $2^rp^s$

Let $p$ be an odd prime and $r,s,m$ be positive integers. In this study, we initiate our exploration by delving into the intricate structure of all repeated-root cyclic codes and their duals with a length of $2^rp^s$ over the finite field $\mathbb{F}_{p^m}$. Through the utilization of CSS and Steane's constructions, a series of new quantum error-correcting (QEC) codes are constructed with parameters distinct from all previous constructions. Furthermore, we provide all maximum distance separable (MDS) cyclic codes of length $2^rp^s$, which are further utilized in the construction of QEC MDS codes. Finally, we introduce a significant number of novel entanglement-assisted quantum error-correcting (EAQEC) codes derived from these repeated-root cyclic codes. Notably, these newly constructed codes exhibit parameters distinct from those of previously known constructions.

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arXiv2024

Binary duadic codes and their related codes with a square-root-like lower bound

Binary cyclic codes have been a hot topic for many years, and significant progress has been made in the study of this types of codes. As is well known, it is hard to construct infinite families of binary cyclic codes [n, n+1/2] with good minimum distance. In this paper, by using the BCH bound on cyclic codes, one of the open problems proposed by Liu et al. about binary cyclic codes (Finite Field Appl 91:102270, 2023) is settled. Specially, we present several families of binary duadic codes with length 2^m-1 and dimension 2^(m-1), and the minimum distances have a square-root-like lower bound. As a by-product, the parameters of their dual codes and extended codes are provided, where the latter are self-dual and doubly-even.

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arXiv2022

Several classes of optimal ternary cyclic codes

Cyclic codes have efficient encoding and decoding algorithms over finite fields, so that they have practical applications in communication systems, consumer electronics and data storage systems. The objective of this paper is to give eight new classes of optimal ternary cyclic codes with parameters $[3^m-1,3^m-1-2m,4]$, according to a result on the non-existence of solutions to a certain equation over $F_{3^m}$. It is worth noticing that some recent conclusions on such optimal ternary cyclic codes are some special cases of our work. More importantly, three of the nine open problems proposed by Ding and Helleseth in [8] are solved completely. In addition, another one among the nine open problems is also promoted.

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arXiv2019

Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, where we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several new families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. Moreover, several new classes of q-ary MDS EAQECCs of length n > q + 1 are also obtained.

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arXiv2016

A Class of Two-Weight and Three-Weight Linear Codes and Their Duals

The objective of this paper is to construct a class of linear codes with two nonzero weights and three nonzero weights by using the general trace functions, which weight distributions has been determined. These linear codes contain some optimal codes, which meets certain bound on linear codes. The dual codes are also studied and proved to be optimal or almost optimal. These codes may have applications in authentication codes, secret sharing schemes and strongly regular graphs.

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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.

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