【摘要】：本文主要研究了兩類有限非鏈環(huán)上的線性碼及其MacWilliamss恒等式,具體內(nèi)容如下：(1)研究了環(huán)R=Z4+vZ4(v2v)上的線性碼及其MacWilliams恒等式.首先給出了環(huán)R上線性碼的Gray映射及其投影映射的性質(zhì),得到了環(huán)R上線性碼與通過投影映射得到的線性碼的極小Lee重量的關(guān)系,然后給出了環(huán)R上線性碼的Gray重量計數(shù)多項式和對稱重量計數(shù)多項式的定義,進(jìn)一步地確定了環(huán)R上線性碼和其對偶碼之間基于Gray重量計數(shù)多項式,對稱重量計數(shù)多項式和Lee重量計數(shù)多項式的MacWilliams恒等式.(2)研究了環(huán)Rk,m=Fq[u,v]/k;,vm,uv-vu上的線性碼及其MacWilliams恒等式,其中q是素數(shù)p的方冪且k≥m≥1.首先定義了Lee重量并給出了Rk,m到Fkmq的Gray映射,此映射關(guān)于Lee重量具有保距性和保對偶性,然后證明了環(huán)Rk,m上線性碼相應(yīng)重量計數(shù)多項式的MacWilliams恒等式,特別地給出了環(huán)Rk,。上線性碼關(guān)于Lee重量計數(shù)多項式的MacWilliams恒等式.
[Abstract]:In this paper, we study two kinds of linear codes and their MacWilliamss identities over finite nonlinked rings. The main contents are as follows: (1) the linear codes and their MacWilliams identities on the ring R=Z4 vZ4 (v2v) are studied. Firstly, the properties of Gray mapping and projection mapping of linear codes over ring R are given, and the relation between linear codes on ring R and minimal Lee weights of linear codes obtained by projection mapping is obtained. Then, the definitions of Gray weight counting polynomial and symmetric weight counting polynomial of linear codes over ring R are given, and the Gray weight counting polynomials between linear codes and their dual codes over ring R are further determined. The MacWilliams identities of symmetric weight counting polynomials and Lee weight counting polynomials. (2) We study the linear codes and their MacWilliams identities on the ring RKN mq [UV] / KV / VMU, where Q is the power of the prime p and k 鈮，

本文編號：2189157