The Nonexistence of Quaternary Linear Codes With Parameters $[243,5,181],[248,5,185]$ and $[240,5,179]$

Abstract

Let $n_4(k,d)$ and $d_4(n,k)$ denote the smallest value of $n$ and the largest value of $d$, respectively, for which there exists an $[n,k,d]$ code over the Galois field $GF(4)$. It is known (cf. Boukliev [1] and Table B.2 in Hamada [6]) that (1) $n_4(5,179)=240$ or $249$, $n_4(5,181)=243$ or $244,n_4(5,182)=244$ or $245,n_4(5,185)=248$ or $249$ and (2) $d_4(240,5)=178$ or $179$ and $d_4(244,5)=181$ or $182$. The purpose of this paper is to prove that (1) $74(5,179)=241,n_4(5,181)=244,n_4(5,182)=245,n_4(5,185)=249$ and (2) $d_4(240,5)=178$ and $d_4(244,5)=181$.