有限域上最优码的一种新的构造方法

基于域F_p^m上一类特殊的矩阵,定义了环R(_p^m,k)=F_p^mu]/k>到F_p^p_mj的一个新的Gray映射,其中u^k=0、p为素数、j为正整数且p^j－1+1≤k≤p^j.得到了环R(_p^m,k)上码长为任意长度N的(1+u)常循环码的Gray象是F_p^m上长为p^jN的保距线性循环码,并给出了Gray象的生成多项式,构造了F₃,F₅和F₇上的一些最优线性循环码.

A New Method on the Construction of OptimaI Codes over Finite FieId

Let R(_p^m, k) denote the polynomial residue ring F_p^mu]/k>, where p^j－1+1≤k≤p^j and u^k=0 for some positive prime number p and positive integer j.Based on a kind of matrix over F_p^m, a new Gray map from R(_p^m, k) to F_p^p_mj is defined.It is proved that the Gray image of a linear (1+u) constacyclic code of an arbitrary length N over R(_p^m, k) is a distance invariant linear cyclic code of length p^jN over F_p^m.Moreover, the generator polynomial of the Gray image of such a constacyclic code is determined, and some optimal linear cyclic codes over F₃, F₅ and F₇ are constructed via the Gray map.