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,$$\text{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].