Group codes over symmetric groups

Abstract

Let $ \Bbb F_{q} $ be a finite field of characteristic $ q $ and $ S_n $ a symmetric group of order $ n! $. In this paper, group codes in the symmetric group algebras $ \Bbb F_{q}S_n $ with $ q > 3 $ and $ n = 3, 4 $ are proposed. We compute the unique (linear and nonlinear) idempotents of $ \Bbb F_q S_n $ corresponding to the characters of symmetric groups and use the results to characterize the minimum distances and dimensions of group codes. Furthermore, we construct MDS group codes and almost MDS group codes in $ \Bbb F_q S_3 $ and $ \Bbb F_q S_4 $.

Keywords

• group codes
• group algebra
• minimum distance
• symmetric group