Some High Rate Linear Codes over $GF(5)$ and $GF(7)$

Let an $[n,k,d]_q$ code be a linear code of length $n$, dimension $k$, and with minimum Hamming distance $d$ over $GF(q)$. The ratio $R=k/n$ is called the rate of a code. In this paper, $[62,53,6]_5$, $[62,48,8]_5$, $[71,56,8]_5$, $[124,113,6]_5$, $[43,36,6]_7$, $[33,23,7]_7$, and $[27,18,7]_7$ high-rate codes and new codes with parameters $[42,14,19]_5$, $[42,15,18]_5$, $[48,13,24]_5$, $[52,12,28]_5$, $[71,15,38]_5$, $[71,16,36]_5$, $[72,12,41]_5$, $[78,10,50]_5$, $[88,11,54]_5$, $[88,13,51]_5$, $[124,14,77]_5$, $[32,12,15]_7$, $[32,10,17]_7$, $[36,10,20]_7$, and $[48,10,29]_7$ are constructed. The codes with parameters $[62,53,6]_5$ and $[43,36,6]_7$ are optimal.