群可分码及其在构造重量为3的最优常复合码中的应用

On the lengths of divisible codes

In this article, the effective lengths of all $q^r$-divisible linear codes over $\mathbb{F}_q$ with a non-negative integer $r$ are determined. For that purpose, the $S_q(r)$-adic expansion of an integer $n$ is introduced. It is shown that there exists a $q^r$-divisible $\mathbb{F}_q$-linear code of effective length $n$ if and only if the leading coefficient of the $S_q(r)$-adic expansion of $n$ is non-negative. Furthermore, the maximum weight of a $q^r$-divisible code of effective length $n$ is at most $σq^r$, where $σ$ denotes the cross-sum of the $S_q(r)$-adic expansion of $n$. This result h…

可整除的极小码

Divisible minimal codes

本文研究极小码的最小长度问题，确定了若干小参数下的精确值并给出完整分类，同时研究了码字重量可被常数整除的极小码，并改进了部分二元线性码的最小距离下界。

一类具有局部性2的自正交可分码

A family of self-orthogonal divisible codes with locality 2

本文利用有限域上的迹和范数函数构造了一族具有局部性2的自正交可分码，通过Gaussian和确定了三种情形下的重量分布，并推导出若干最优线性码及局部可恢复码，可应用于格构造与分布式存储。

Determination of sizes of optimal three-dimensional optical orthogonal codes of weight three with the AM-OPP restriction

In this paper, we further investigate the constructions on three-dimensional $(u\times v\times w,k,1)$ optical orthogonal codes with the at most one optical pulse per wavelength/time plane restriction (briefly AM-OPP $3$-D $(u\times v\times w,k,1)$-OOCs) by way of the corresponding designs. Several new auxiliary designs such as incomplete holey group divisible designs and incomplete group divisible packings are introduced and therefore new constructions are presented. As a consequence, the exact number of codewords of an optimal AM-OPP $3$-D $(u\times v\times w,3,1)$-OOC is finally determined …

Divisible Codes for Quantum Computation

Divisible codes are defined by the property that codeword weights share a common divisor greater than one. They are used to design signals for communications and sensing, and this paper explores how they can be used to protect quantum information as it is transformed by logical gates. Given a CSS code $\mathcal{C}$, we derive conditions that are both necessary and sufficient for a transversal diagonal physical operator $U_Z$ to preserve $\mathcal{C}$ and induce $U_L$. The group of $Z$-stabilizers in a CSS code $\mathcal{C}$ is determined by the dual of a classical $[n, k_1]$ binary code $\math…

Quasi optimal anticodes: structure and invariants

It is well-known that the dimension of optimal anticodes in the rank-metric is divisible by the maximum m between the number of rows and columns of the matrices. Moreover, for a fixed k divisible by m, optimal rank-metric anticodes are the codes with least maximum rank, among those of dimension k. In this paper, we study the family of rank-metric codes whose dimension is not divisible by m and whose maximum rank is the least possible for codes of that dimension, according to the Anticode bound. As these are not optimal anticodes, we call them quasi optimal anticodes (qOACs). In addition, we ca…

线性长度最优q元等重码和等复合码

Linear Size Optimal q-ary Constant-Weight Codes and Constant-Composition Codes

一个最优的等重码$w$具有线性长度当且仅当它的距离$d$至少为$2w-1$。当$d\geq 2w$时，确定这种恒定组成或恒定重量代码的确切大小是微不足道的，但是$d=2w-1$的情况先前仅针对二进制和三进制恒定组成和恒定重量代码以及针对一些零星的实例而被解决。基于差三角集的一种新的推广，给出了权$w$和距离$2w-1$的拟循环最优常合成常重码的构造.结果，对于所有这样的长度足够大的码，确定了权重$w$和距离$2w-1$的最佳恒定组成码和最佳恒定权重码的大小。这解决了Etzion的一个开放问题。除了两种情况外，还确定了所有$w\leq 6$的重量$w$和距离$2w-1$的最佳常数组合码的大小。

关于不可约循环码重量分布的一个注记

A Note on Weight Distributions of Irreducible Cyclic Codes

通常，确定不可约循环码的重量分布是困难的。本文讨论了一个不可约循环码具有最大数目的不同非零权的情形，并给出了一个充分必要条件。在这种情况下，我们还获得了码字的权重的可除性。进一步，我们给出了只有一个非零权的不可约循环码的一个充要条件。最后，我们确定了某些情况下不可约循环码的重量分布。

Optimal codes with small constant weight in $\ell_1$-metric

Motivated by the duplication-correcting problem for data storage in live DNA, we study the construction of constant-weight codes in $\ell_1$-metric. By using packings and group divisible designs in combinatorial design theory, we give constructions of optimal codes over non-negative integers and optimal ternary codes with $\ell_1$-weight $w\leq 4$ for all possible distances. In general, we derive the size of the largest ternary code with constant weight $w$ and distance $2w-2$ for sufficiently large length $n$ satisfying $n\equiv 1,w,-w+2,-2w+3\pmod{w(w-1)}$.