论文标题
关于Galois自助式GRS和TGRS代码的存在
On the Existence of Galois Self-Dual GRS and TGRS Codes
论文作者
论文摘要
令$ q = p^m $为主要功率，$ e $是$ 0 \ leq e \ leq m-1 $的整数。 $ e $ -galois的自助式代码是Euclidean $（e = 0）$和Hermitian（$ e = \ frac {m} {2} $，带有$ M $）的概括。在本文中，对于线性代码$ \ c $和非零矢量$ \ bm {u} \ in \ f_q^n $，我们为双扩展代码$ \ useverline {\ c} {\ c} [\ bm {u}] $ $ \ c $ $ e $ e $ e $ - galois sefors of be y n of tline {\ c} [\ bm {u}] $。由此，提出了一种新的系统方法，以证明存在$ e $ -galois自动偶数代码。通过这种方法，我们证明$ e $ - galois自dual（扩展）广义的芦苇 - 固体（grs）代码为长度$ n> \ min \ {p^e+1，p^{m-e} +1 \} $不存在，在$ 1 \ leq e \ leq e \ leq m-1 $的位置。此外，基于扭曲GRS（TGRS）代码的非GRS属性，我们表明在许多情况下，$ e $ galois self-dual（扩展）TGRS代码不存在。此外，我们为$（\ ast）$ -TGRS代码提供了足够且必要的条件，使其成为Hermitian self-dual，然后构建几个新的Hermitian self-Dual $（+）$ - tgrs和$（\ ast）$ -TGRS代码。
Let $q=p^m$ be a prime power and $e$ be an integer with $0\leq e\leq m-1$. $e$-Galois self-dual codes are generalizations of Euclidean $(e=0)$ and Hermitian ($e=\frac{m}{2}$ with even $m$) self-dual codes. In this paper, for a linear code $\C$ and a nonzero vector $\bm{u}\in \F_q^n$, we give a sufficient and necessary condition for the dual extended code $\underline{\C}[\bm{u}]$ of $\C$ to be $e$-Galois self-orthogonal. From this, a new systematic approach is proposed to prove the existence of $e$-Galois self-dual codes. By this method, we prove that $e$-Galois self-dual (extended) generalized Reed-Solomon (GRS) codes of length $n>\min\{p^e+1,p^{m-e}+1\}$ do not exist, where $1\leq e\leq m-1$. Moreover, based on the non-GRS properties of twisted GRS (TGRS) codes, we show that in many cases $e$-Galois self-dual (extended) TGRS codes do not exist. Furthermore, we present a sufficient and necessary condition for $(\ast)$-TGRS codes to be Hermitian self-dual, and then construct several new classes of Hermitian self-dual $(+)$-TGRS and $(\ast)$-TGRS codes.
