Construction of Galois self-orthogonal MDS codes with larger dimensions

Abstract

Let $q=p^m$ be a prime power, e be an integer with $0\le e\le m-1$ and $s=\gcd (e,m)$. In this paper, for a vector $v\in {\left(F_q^{⁎}\right)}^n$ and a q-ary linear code $C$, we give some necessary and sufficient conditions for the equivalent code ${\Phi }_v\left(C\right)$ of $C$ and the extended code of ${\Phi }_v\left(C\right)$ to be e-Galois self-orthogonal. We then directly obtain some necessary and sufficient conditions for (extended) generalized Reed-Solomon (GRS and EGRS) codes to be e-Galois self-orthogonal. From this we show that if $k\ge \min \{\max \{p^e,\lceil \frac{n+p^e}{p^e+1}\rceil \},\max \{p^{m-e},\lceil \frac{n+p^{m-e}}{p^{m-e}+1}\rceil \left\}\right\}$, there is no ${[n,k]}_q$ e-Galois self-orthogonal (extended) GRS code. Furthermore, for all possible e satisfying $0\le e\le m-1$, we classify them into three cases (1) $\frac{m}{s}$ odd and p even; (2) $\frac{m}{s}$ odd and p odd; (3) $\frac{m}{s}$ even, and construct several new classes of e-Galois self-orthogonal maximum distance separable (MDS) codes. It is worth noting that our e-Galois self-orthogonal MDS codes can have dimensions greater than $\lfloor \frac{n+p^e-1}{p^e+1}\rfloor$, which are not covered by previously known ones. Moreover, by propagation rules, we obtain some new MDS codes with Galois hulls of arbitrary dimensions.