环上线性码的深度谱以及深度分布的一个注记

摘要

给出了环R1=Fq[u]/(um)(um=0,m≥2,q为素数的方幂)以及环R2=Fq+ uFq+ vFq+ uvFq上码长为n的线性码C的深度谱以及深度分布计算公式,证明了环Ri(i=1,2)上n维向量的深度值均可取遍0,1,…,n,以及对任意A={l1,l2,…,lk}∈{0,1,…,n},均存在Ri(i=1,2)上码长为n的线性码C,其深度谱恰为A.%Let q be a power of the prime p and Fq be the q elements finite field. For tow residue rings ft, = Fq[u]/(um) ( um =0, m≥2 ) and R2 = Fq + uFq + vFq + unFq , the formula for the depth distribution of a linear code over Ri ( i = 1,2) is obtained. It is also proved that for any subset A ε{0,1 , ···, n } there exists a linear code over Ri ( i = 1,2) with the depth spectrum A.