LCD codes and self-orthogonal codes in generalized dihedral group algebras
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LCD codes and self-orthogonal codes in generalized dihedral group algebras Yanyan Gao1,2 · Qin Yue1,3
· Yansheng Wu1,3
Received: 28 September 2019 / Revised: 11 June 2020 / Accepted: 13 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let Fq be a finite field with q elements, D2n, r a generalized dihedral group with gcd(2n, q) = 1, and Fq [D2n, r ] a generalized dihedral group algebra. Firstly, an explicit expression for primitive idempotents of Fq [D2n, r ] is determined, which extends the results of Brochero Martínez (Finite Fields Appl 35:204–214, 2015). Secondly, all linear complementary dual (LCD) codes and self-orthogonal codes in Fq [D2n, r ] are precisely described and counted. Some numerical examples are also presented to illustrate our main results. Keywords Group algebra · Generalized dihedral group · LCD codes · Self-orthogonal codes Mathematics Subject Classification 94B05 · 11T71 · 16S34
1 Introduction Group algebra is one of the important sources of constructing linear codes. We call C a group code if C is just a right ideal in a group ring R[G], where R is a commutative ring and G is a finite group. In particular, if G is abelian, then C is an abelian code. A brief survey on group codes of some recent results is provided as follows.
Communicated by D. Ghinelli.
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Qin Yue [email protected] Yanyan Gao [email protected] Yansheng Wu [email protected]
1
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211100, People’s Republic of China
2
Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing 211167, People’s Republic of China
3
State Key Laboratory of Cryptology, P. O. Box 5159, Beijing 100878, People’s Republic of China
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Y. Gao et al.
(1) Ferraz et al. [10] determined the number of simple components of a semisimple finite abelian group algebra, in term of the number of q-cyclotomic classes. (2) Brochero Martínez et al. [3] determined an explicit expression for the primitive idempotents of Fq [G], where Fq is a finite field, G is a finite cyclic group of order p k , and p is an odd prime with gcd(q, p) = 1. Brochero Martínez [2] also showed explicitly all central irreducible idempotents and their Wedderburn decomposition of the dihedral group algebra Fq [D2n ] if every prime divisor of n divides q − 1. (3) Polcino Milies et al. [21] calculated the minimum distances and the dimensions of all cyclic codes of length p n over a finite field Fq , if p is an odd prime, Fq is a finite field with q elements, and q generates the group of invertible elements of the residue ring module p n , denoted by Z pn . (4) Jitman et al. [14] gave a characterization and an enumeration of Euclidean self-dual and Euclidean self-orthogonal abelian codes in principal ideal group algebras. Then Jitman et al. continued this work and studied the Hermitian self-dual abelian codes in a group ring Fq 2 [G] in [15]. (5) Choosuwan et al. [8] gave the complete enumeration of self-dual abelian codes in nonprincipal ideal gr
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