New Minimum Distance Bounds for Linear Codes over Small Fields

Let $[n,k,d]_q$-codes be linear codes of length $n$, dimension $k$, and minimum Hamming distance $d$ over $GF(q)$. In this paper we consider codes over $GF(3)$, $GF(5)$, $GF(7)$, and $GF(8)$. Over $GF(3)$, three new linear codes are constructed. Over $GF(5)$, eight new linear codes are constructed and the nonexistence of six codes is proved. Over $GF(7)$, the existence of 33 new codes is proved. Over $GF(8)$, the existence of ten new codes and the nonexistence of six codes is proved. All of these results improve the corresponding lower and upper bounds in Brouwer's table [1].