四川大学学报（自然科学版）2024，Vol.61Issue(2)：19-22,4.DOI:10.19907/j.0490-6756.2024.021002
求MDS码权多项式的组合方法
A combinatorial method for calculating the weight polynomial of MDS codes
摘要
Abstract
MDS code is a good code and satisfy the Singleton bound.Nowadays,MDS code has been widely used in practice due to its excellent decoding ability.The weight polynomial of MDS code is completely deter-mined by its parameters[n,k,d].In this paper,by using the inclusion-exclusion principle,we calculate the number of code words with fixed Hamming weights in MDS code and gives a new proof of MDS code weight polynomial.Let d≤w≤n and choose d positions from n positions to form set S.We prove that the number of code words in the MDS code with support set S and element 1 at the first position of S is ∑w-dj=0(-1)j(w-1j)qw-d-j.The key of proof is to apply the inclusion-exclusion principle for the set of code words with support set contained in S and element 1 at the first position of S,and to use the linear independence of any d-1 columns in the MDS code parity check matrix.The proof intuitively reveals the combinatorial meaning of the coefficients in MDS code weight polynomial.In contrast to the proof in textbook,our proof does not use the MacWilliams identity.关键词
MDS 码/权多项式/容斥原理Key words
MDS code/Weight polynomial/Inclusion-exclusion principle分类
数理科学引用本文复制引用
李岩,孙久兴..求MDS码权多项式的组合方法[J].四川大学学报（自然科学版）,2024,61(2):19-22,4.基金项目
密码科学技术国家重点实验室开放课题(MMKFKT201910) （MMKFKT201910）
