环F p+u F p+v F p+uv F p+v 2 F p+uv 2 F p上的二次剩余码

摘要

设R=F p+uF p+vF p+uvF p+v 2 F p+uv 2 F p,其中u 2=1,v 3=v,p是一个奇素数.研究了环R上素长度n=q的二次剩余码,其中q(q≠p)是一个奇素数且p是模q的二次剩余.首先研究了环R上长度为n的循环码,根据其幂等生成元定义了环R上的二次剩余码,进一步讨论了该环上二次剩余码与其扩展码的关系.最后,为了验证结果的正确性,给出了F p+uF p+vF p+uvF p+v 2 F p+uv 2 F p上二次剩余码的幂等生成元的两种具体形式.%Let R =F p +uF p +vF p +uvF p +v 2 F p +uv 2 F p ,where u 2 =1,v 3 =v ,and p is an odd prime.Quadratic residue codes of prime length n =q over the ring R was investigated,where q (q ≠p )is an odd prime such that p is a quadratic residue modulo q .The cyclic codes of length n over R were studied,and then the quadratic residue codes over R in terms of idempotent generators were difined.Moreover,the relation between these codes and their extended codes are discussed.Finally,two specific forms of idempotent generators of quadratic residue codes over F p +u F p +v F p +uvF p +v 2 F p +uv 2 F p were given to illustrate some results.