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

Abstract

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,and  [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].