关于指数为2和4的自对偶负循环码
On self-dual negacirculant codes of index two and four
摘要：
本文研究了当$n=2p,$与$p\equiv 3~({\rm mod}~4)$和$p$是素数时，$x^n+1$在$\mathbb{F}_q, $上的一种特殊分解，其中$q$是素数幂$\equiv 3~({\rm mod}~4)$.给定这样一个$q$，存在无穷多个这样的$p$，它们通过算术级数中的Artin猜想承认$q$为原根.这个数论猜想已知在GRH下成立。我们研究了双重（resp。本文给出了有限域$n$上的（4）-双循环码，包括自对偶子类的精确计数，以及它所含码的相对距离的一个修正的Varshamov-Gilbert界.
英文摘要：
In this paper, we study a special kind of factorization of $x^n+1$ over $\mathbb{F}_q, $ with $q$ a prime power $\equiv 3~({\rm mod}~4)$ when $n=2p,$ with $p\equiv 3~({\rm mod}~4)$ and $p$ is a prime. Given such a $q$ infinitely many such $p$'s exist that admit $q$ as a primitive root by the Artin conjecture in arithmetic progressions. This number theory conjecture is known to hold under GRH. We study the double (resp. four)-negacirculant codes over finite fields $\mathbb{F}_q, $ of co-index such $n$'s, including the exact enumeration of the self-dual subclass, and a modified Varshamov-Gilbert bound on the relative distance of the codes it contains.
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