环R_(k,m)上的线性码及其MacWilliams恒等式

Linear Codes and Their MacWilliams Identity Over the Ring R_(k,m)

研究了环R_(k,m)=Fq[u,v]/〈uk,vm,uv-vu〉上的线性码及其MacWilliams恒等式,其中q是素数p的方幂且k≥m≥1.首先给出了R_(k,m)到Fkmq的Gray映射,此映射关于Lee重量具有保距性和保对偶性,然后证明了环R_(k,m)上线性码相应重量计数多项式的MacWilliams恒等式,特别地给出了环R_(k,m)上线性码关于Lee重量计数多项式的MacWilliams恒等式.

We study linear codes and their MacWilliams identity over the ring R k,m=F q[u,v]/〈u k,v m,uv-vu〉,which q is a power of the prime p and k and m are both positive integers.First we define the Lee weights of the elements on R k,m and introduce a distance and duality preserving Gray map from R k,m to F km q together with the Lee weight.Then we give the MacWilliams identities for codes over R k,m for all the relevant weight enumerators.Especially,the MacWilliams identities between linear codes and their dual over R k,m with respect to Lee weight enumerator are obtained.