On the minimum length of q-ary linear codes of dimension four

Abstract

Values and lower bounds for n_q(4,d) for general q are given, where n_q(k,d) denotes the minimum integer n for which there exists a linear code of length n, dimension k and minimum Hamming distance d over the Galois field GF(q). As a result for the nonprojective linear codes, we prove the nonexistence of an [n,4,2q³−rq²−q+1]_q code attaining the Griesmer bound for $\text{q>r,}\text{r=3,4,}$ and for $\text{q>2(r−1),}\text{r⩾3}$.