线性码的Hull的研究

线性码的Hull是指它与其对偶码的交集，是近年来纠错码领域的一个研究热点。Hull维数越低则确定线性码的自同构群的算法复杂度越低，Hull维数为0的LCD码在密码学中有着重要应用，自对偶码在编码、密码、组合等学科都有重要应用。本项目主要利用代数、数论、组合等工具辅以计算机搜索，研究Hull维数为1的线性码、LCD码和自对偶码的构造和重量分布问题，具体内容如下：1）根据GRS码（扩展的GRS码）的特点，研究它们Hull的特征，构造出一些新参数的Hull维数达到最大值的MDS自对偶码、LCD码和Hull维数为1的码；2）分析Hull维数为1的BCH码的生成多项式以及线性码的生成矩阵的特点，构造出一些Hull维数为1的码，并利用数论或组合工具计算其重量分布；3）构造一些校验多项式的零点数较少的二元、三元LCD BCH码，利用指数和等工具计算它们的重量分布。

The hull of a linear code refers to its intersection with its dual code, which is a research hotspot in the field of error-correcting codes in recent years. The lower the hull dimension is, the lower the algorithm complexity is to determine the automorphism group of linear codes. LCD codes with hull dimension 0 have important applications in cryptography, and self-dual codes have important applications in coding, cryptography, combinatorics and other disciplines. In this project we mainly use the tools of algebra, number theory, combinatorics, supplemented by computer search, to investigate the construction of linear codes with hull dimension 1, LCD codes and self-dual codes and their weight distributions. The details are described as follows: 1) investigate the properties of GRS codes (Extended GRS codes) with hull dimensions 1, 0 and half of the code length, search some optimal linear codes with the properties, construct some optimal linear codes with new parameters; 2) investigate the properties of BCH codes with hull dimension 1 and the generator matrix of linear codes with hull dimension 1, construct linear codes with the properties, and calculate their weight distributions; 3）construct some binary and ternary LCD BCH codes, whose parity polynomials have less zeros, study their weight distribution by using exponential sums and other tools.