On the minimum length of some linear codes

Abstract

We determine the minimum length n _q (k, d) for some linear codes with k ≥ 5 and q ≥ 3. We prove that n _q (k, d) = g _q (k, d) + 1 for $q^{k-1}-2q^{\frac{k-1}{2}}-q+1 \le d \le q^{k-1}-2q^{\frac{k-1}{2}}$ when k is odd, for $q^{k-1}-q^{\frac{k}{2}}-q^{{\frac{k}{2}}-1} -q+1 \le d \le q^{k-1}-q^{\frac{k}{2}}-q^{{\frac{k}{2}}-1}$ when k is even, and for $2q^{k-1}-2q^{k-2}-q^2-q+1 \le d \le 2q^{k-1}-2q^{k-2}-q^2$.