Self-Dual Cyclic Codes over M2 (F2+uF2)

Abstract

Self-dual cyclic codes (Mathematics Subject Classification, 2000: 94B15）form an important class of linear codes due to their significance in coding theory and decoding theory. Recently, there are some papers on cyclic codes over rings, these codes caught the attention of researchers. A. Hammons et al. (1994; 301-319) studied some results on codes, and they have shown a relationship between non-linear binary codes and linear codes. Linear codes over the matrix ring have been studied in the survey of F. Oggier et al. (2012: 734-746). The advantage of matrix rings is that they are non-commutative. On structures of cyclic codes and their dual codes over non-commutative finite rings forms an important and new topic in coding theory in the works of modern scientists within the field (Luo and Parampalli, 2018: 1109-1117; Bhowmick et al., 2018; Alahmadi et al., 2013: 2837-2847; Pal et al., 2019). R. Luo and U. Parampalli (2018: 1109-1117) studied cyclic codes over . Some optimal cyclic codes over  were obtained. S. Bhowmick, S. Bagchi, R.K. Bandi (2018) studied the structures of the ring and then focused on algebraic structures of cyclic codes and self-dual cyclic codes over. A. Alahmadi, H. Sboui, P. Sol´e, O. Yemen (2013) characterized cyclic codes and self-dual cyclic codes over the matrix ring . J. Pal, S. Bhowmick, B. Satya (2019) studied some results on cyclic codes over .

Motivated by R. Luo and U. Parampalli (2018: 1109-1117) and Bhowmick et al., (2018), in this paper, we study self-dual cyclic codes over the matrix ring . The rest of this paper is organized as follows. In section 2, we review some results on the matrix ring, give a Gray map from this ring to. In section 3, we study the dual codes of cyclic codes over . A necessary and sufficient condition for cyclic codes to be self-dual is given. As an application, some self-dual codes over  are obtained by the Gray map.