A Recursive Method for Determining the One-Dimensional Submodules of Laurent-Ore Modules

We present a method for determining the one-dimensional submodules of a Laurent-Ore module. The method is based on a correspondence between hyperexponential solutions of associated systems and one-dimensional submodules. The hyperexponential solutions are computed recursively by solving a sequence of first-order ordinary matrix equations. As the recursion proceeds, the matrix equations will have constant coefficients with respect to the operators that have been considered.Comment: To appear in the Proceedings of ISSAC 200

Three classes of new optimal cyclic (r,δ)(r,\delta) locally recoverable codes

An $(r, \delta)$-locally repairable code ($(r, \delta)$-LRC for short) was introduced by Prakash et al. for tolerating multiple failed nodes in distributed storage systems, and has garnered significant interest among researchers. An $(r,\delta)$-LRC is called an optimal code if its parameters achieve the Singleton-like bound. In this paper, we construct three classes of $q$-ary optimal cyclic $(r,\delta)$-LRCs with new parameters by investigating the defining sets of cyclic codes. Our results generalize the related work of \cite{Chen2022,Qian2020}, and the obtained optimal cyclic $(r, \delta)$-LRCs have flexible parameters. A lot of numerical examples of optimal cyclic $(r, \delta)$-LRCs are given to show that our constructions are capable of generating new optimal cyclic $(r, \delta)$-LRCs

Two classes of reducible cyclic codes with large minimum symbol-pair distances

The high-density data storage technology aims to design high-capacity storage at a relatively low cost. In order to achieve this goal, symbol-pair codes were proposed by Cassuto and Blaum \cite{CB10,CB11} to handle channels that output pairs of overlapping symbols. Such a channel is called symbol-pair read channel, which introduce new concept called symbol-pair weight and minimum symbol-pair distance. In this paper, we consider the parameters of two classes of reducible cyclic codes under the symbol-pair metric. Based on the theory of cyclotomic numbers and Gaussian period over finite fields, we show the possible symbol-pair weights of these codes. Their minimum symbol-pair distances are twice the minimum Hamming distances under some conditions. Moreover, we obtain some three symbol-pair weight codes and determine their symbol-pair weight distribution. A class of MDS symbol-pair codes is also established. Among other results, we determine the values of some generalized cyclotomic numbers

The Weight Hierarchies of Linear Codes from Simplicial Complexes

The study of the generalized Hamming weight of linear codes is a significant research topic in coding theory as it conveys the structural information of the codes and determines their performance in various applications. However, determining the generalized Hamming weights of linear codes, especially the weight hierarchy, is generally challenging. In this paper, we investigate the generalized Hamming weights of a class of linear code \C over \bF_q, which is constructed from defining sets. These defining sets are either special simplicial complexes or their complements in \bF_q^m. We determine the complete weight hierarchies of these codes by analyzing the maximum or minimum intersection of certain simplicial complexes and all $r$-dimensional subspaces of \bF_q^m, where 1\leq r\leq {\rm dim}_{\bF_q}(\C)

A Family of pp-ary Binomial Bent Functions

For a prime $p$ with $p\equiv 3\,({\rm mod}\, 4)$ and an odd number $m$, the Bentness of the $p$-ary binomial function $f_{a,b}(x)={\rm Tr}_{1}^n(ax^{p^m-1})+{\rm Tr}_{1}^2(bx^{\frac{p^n-1}{4}})$ is characterized, where $n=2m$, a\in \bF_{p^n}^*, and b\in \bF_{p^2}^*. The necessary and sufficient conditions of $f_{a,b}(x)$ being Bent are established respectively by an exponential sum and two sequences related to $a$ and $b$. For the special case of $p=3$, we further characterize the Bentness of the ternary function $f_{a,b}(x)$ by the Hamming weight of a sequence