2019/09/07 by Parinyawat Choosuwan, Choosuwan, Parinyawat, Somphong Jitman +1
Computer Science · Engineering · #94B05 #94B15 #94B60 #Coding theory and cryptography #Cooperative Communication and Network Coding #FOS: Computer and information sciences #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1909.03174
openalex publication_date 2019/09/07 · openalex created_date 2019/09/12 · openalex updated_date 2026/07/28
Self-dual codes over finite fields and over some finite rings have been of interest and extensively studied due to their nice algebraic structures and wide applications. Recently, characterization and enumeration of Euclidean self-dual linear codes over the ring~\mathbbFq+u\mathbbFq+u2\mathbbFq with u3=0 have been established. In this paper, Hermitian self-dual linear codes over \mathbbFq+u\mathbbFq+u2\mathbbFq are studied for all square prime powers~q. Complete characterization and enumeration of such codes are given. Subsequently, algebraic characterization of H-quasi-abelian codes in \mathbbFq[G] is studied, where H≤ G are finite abelian groups and \mathbbFq[H] is a principal ideal group algebra. General characterization and enumeration of H-quasi-abelian codes and self-dual H-quasi-abelian codes in \mathbbFq[G] are given. For the special case where the field characteristic is 3, an explicit formula for the number of self-dual A× ℤ3-quasi-abelian codes in \mathbbF3m[A× ℤ3× B] is determined for all finite abelian groups A and B such that 3\nmid |A| as well as their construction. Precisely, such codes can be represented in terms of linear codes and self-dual linear codes over \mathbbF3m+u\mathbbF3m+u2\mathbbF3m.