New Linear Codes of Dimensions $5$ and $6$ over $GF(8)$

Abstract

Let $[n,k,d;g]$-codes be linear codes of length $n$, dimension $k$ and minimum Hamming distance $d$ over $\mathrm{G}\mathrm{F}(g)$. Let $d_8(n,k)$ be the maximum possible minimum Hamming distance of a linear $[n,k,d;8]$-code for given values of $n$ and $k$. In this paper, twenty-two new linear codes over $\mathrm{G}\mathrm{F}(8)$ are constructed which improve the bounds on $d_8(n,k)$.