The poset metrics that allow binary codes of codimension m to be m-, (m-1)-, or (m-2)-perfect

Computing the poset of layers of a toric arrangement

A toric arrangement is an arrangement of subtori of codimension one in a real or complex torus. The poset of layers is the set of connected components of non-empty intersections of these subtori, partially ordered by reverse inclusion. In this note we present an algorithm that computes this poset in the central case.

Perfect Codes for Uniform Chains Poset Metrics

The class of poset metrics is very large and contains some interesting families of metrics. A family of metrics, based on posets which are formed from disjoint chains which have the same size, is examined. A necessary and sufficient condition, for the existence of perfect single-error-correcting codes for such poset metrics, is proved.

The weighted poset metrics and directed graph metrics

Etzion et al. introduced metrics on $\mathbb{F}_2^n$ based on directed graphs on $n$ vertices and developed some basic coding theory on directed graph metric spaces. In this paper, we consider the problem of classifying directed graphs which admit the extended Hamming codes to be a perfect code. We first consider weighted poset metrics as a natural generalization of poset metrics and investigate interrelation between weighted poset metrics and directed graph based metrics. In the next, we classify weighted posets on a set with eight elements and directed graphs on eight vertices which admit th…

改变小余维数仿射子空间上的几乎完全非线性函数

Changing almost perfect nonlinear functions on affine subspaces of small codimensions

针对 APN 函数，本文研究其代数分解与二次构造，给出修改后仍为 APN 的精确判别准则，并证明所得函数可与原函数扩展仿射不等价。

Smoothness and singularities of the perfect form and the second Voronoi compactification of ${\mathcal A}_g$

We study the cones in the first Voronoi or perfect cone decomposition of quadratic forms with respect to the question which of these cones are basic or simplicial. As a consequence we deduce that the singular locus of the moduli stack ${\mathcal A_g^{\mathop{Perf}}}$, the toroidal compactification of the moduli space of principally polarized abelian varieties of dimension $g$ given by this decomposition, has codimension $10$ if $g \geq 4$. Moreover we describe the non-simplicial locus in codimension $10$. We also show that the second Voronoi compactification ${\mathcal A_{g}^{\mathop{Vor}}}$ h…

加权坐标偏序集块空间的权重分布与Singleton界

Weight Distribution of the Weighted Coordinates Poset Block Space and Singleton Bound

本文确定了加权坐标偏序集块空间的完整权重分布，涵盖多种特例，并研究了$I$-完美码、$t$-完美码与MDS码的关系及对偶定理，分析了链偏序集下$r$-球中码字分布。