矩阵环上的零差分平衡函数  

Zero-difference Balanced Functions on Matrix Rings

作　　者：易宗向[1,2] 余玉银[1] YI Zong-Xiang;YU Yu-Yin(School of Mathematics and Information Science,Guangzhou University,Guangzhou 510006,China;Guangdong University of Science and Technology,Dongguan 523083,China)

机构地区：[1]广州大学数学与信息科学学院,广州510006 [2]广东科技学院,东莞523083

出　　处：《密码学报》2019年第2期165-176,共12页Journal of Cryptologic Research

基　　金：国家自然科学基金(61502113);广东省自然科学基金(2015A030310174)~~

摘　　要：零差分平衡函数在编码和组合设计上有很多应用,例如可以构造常重复合码、常重码、集合差系统、跳频序列等.当零差分平衡函数满足某些条件的时候,这些对象可以达到各自对应的最优界.已有的一些基于分圆陪集构造的零差分平衡函数都是在交换环上的,因此本文考虑在有限域F_q的矩阵环上M_n(q)构造零差分平衡函数.为此,本文首先给出一种使用分圆陪集来构造零差分平衡函数的方法.为了使用这种方法来构造零差分平衡函数,本文证明了对于满足特定条件的可逆矩阵A,其乘法阶r满足r|q^n-1.在此基础上2,本文构造了参数为(q^n,(q^n^2-1)/r+1, r-1)的零差分平衡函数.最后我们具体介绍如何使用本文得到的零差分平衡函数来构造重复合码、常重码和集合差系统等编码和密码学组件.这些组件在给定的约束条件下都可以达到对应的最优界,具有重要的应用价值.Zero-difference balanced(ZDB)functions have many applications in codes and designs,such as the constructions of constant composition codes,constant weight codes,difference systems of sets,and frequency-hopping sequences.Those constructions can meet some optimal bounds if the ZDB functions satisfy some conditions.Most of known ZDB functions are constructed based on cyclotomic cosets over communicative rings.Hence ZDB functions on matrix rings M_n(q)over finite field F_q,are considered in this study.This paper proves that the invertible matrices satisfying certain condition must have its multiplicative order r meeting the condition r|q^n-1.Based on this result,new ZDB functions on matrix rings with parameters(q^n^2,(q^n^2-1)/r+1,r-1),are proposed.Finally,some frameworks of applications of ZDB function are presented,such as the constructions of constant composition codes,constant weight codes,and difference systems of sets.The proposed constructions meet some optimal bounds under some constraints and have many important applications.

关 键 词：常用复合码 常重码 集合差系统 跳频序列 零差分平衡函数 矩阵环 

分 类 号：O157.4[理学—数学]