On Optimal Linear Codes over ${\mathbb{F}}_8$

Abstract

Let $n_q(k,d)$ be the smallest integer $n$ for which there exists an $[n,k,d]_q$ code for given $q,k,d$. It is known that $n_8(4,d) = \sum_{i=0}^{3} \left\lceil d/8^i \right\rceil$ for all $d \ge 833$. As a continuation of Jones et al. [Electronic J. Combinatorics 13 (2006), #R43], we determine $n_8(4,d)$ for 117 values of $d$ with $113 \le d \le 832$ and give upper and lower bounds on $n_8(4,d)$ for other $d$ using geometric methods and some extension theorems for linear codes.