New Binary Quantum Codes from Group Rings and Skew Group Rings

In this paper, we consider group rings and skew group rings and use them to construct binary quantum error-correcting codes(QECCs, for short). We study three classes of groups: direct product of two cyclic groups, dihedral group, direct product of cyclic group and dihedral group. And several special groups. Using these groups we can get construction matrix of the linear codes over \(\mathbb {F}_4\) ${\mathbb{F}}_4$ . Then by computer search, we get 12 new binary QECCs with parameters [[40,16,7]], [[42,14,8]], [[48,20,8]], [[50,24,7]], [[54,34,6]], [[55,21,9]], [[60,38,6]], [[60,32,7]], [[70,42,7]], [[72,50,6]], [[75,47,7]], [[75,39,8]]. They both break the best-known lower bound on minimum distance in the Grassl’s code tables.