New \(6\)-Dimensional Linear Codes over \(GF(8)\) and \(GF(9)\)

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, the existence of the following codes is proven:

\[[42, 6, 30]_8, [49, 6, 36]_8, [78, 6, 60]_8, [84, 6, 65]_8, [91, 6, 71]_8, [96, 6, 75]_8, [102, 6, 80]_8, [108, 6, 85]_8, [114, 6, 90]_8,
\]

\[\text{and} \quad [48, 6, 35]_9, [54, 6, 40]_9, [60, 6, 45]_9, [96, 6, 75]_9, [102, 6, 81]_9, [108, 6, 85]_9, [114, 6, 90]_9, [126, 6, 100]_9, [132, 6, 105]_9.
\]

The nonexistence of five codes over \( GF(9) \) is also proven. All of these results improve the respective upper and lower bounds in Brouwer’s table [2].