Construction of new linear codes with good parameters from group rings and skew group rings

Abstract

In this paper, we use the left principal ideals of group rings and skew group rings to construct linear codes over small finite fields. We study three class of groups: Semidirect product of two cyclic groups, direct product of a cyclic group and semidirect product of two cyclic groups, wreath product of a cyclic group of order n and a cyclic group of order 2. Using these groups, we can get some generator matrices over $F_q$. Then by computer search, we obtain 18 new linear codes with parameters ${[32,19,8]}_3$, ${[36,22,8]}_3$, ${[39,21,10]}_3$,${[40,26,8]}_3$,${[40,30,6]}_3$, ${[55,16,22]}_3$, ${[40,8,24]}_4$, ${[40,11,20]}_4$, ${[40,27,8]}_4$, ${[42,29,8]}_4$, ${[42,25,10]}_4$,${[50,13,24]}_4$, ${[50,9,30]}_5$, ${[39,12,19]}_5$, ${[40,8,26]}_7$, ${[18,12,6]}_9$, ${[21,16,5]}_9$, ${[42,7,30]}_9$.