Geodesic
Self-Dual Cyclic Codes Over $M_2(ℤ_4)$
Discussiones Mathematicae. General Algebra and Applications, Tome 42 (2022) no. 2, pp. 349-362

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In this paper, we study the structure of cyclic codes over M_2(ℤ_4) (the matrix ring of matrices of order 2 over ℤ_4), which is perhaps the first time that the ring is considered as a code alphabet. This ring is isomorphic to ℤ_4 [ w ] + U ℤ_4 [w], where w is a root of the irreducible polynomial x^2 + x + 1 ∈ℤ_2 [ x ] and U ≅[ 1 amp; 1; 1 amp; 1 ]. In our work, we first discuss the structure of the ring M_2(ℤ_4) and then focus on the structure of cyclic codes and self-dual cyclic codes over M_2(ℤ_4). Thereafter, we obtain the generators of the cyclic codes and their dual codes. A few non-trivial examples are given at the end of the paper.
Keywords: codes over $\mathbb{Z}_4 + u\mathbb{Z}_4$, Gray map, Lee weight, self-dual codes 
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Bhowmick, Sanjit; Pal, Joydeb; Bandi, Ramakrishna; Bagchi, Satya. Self-Dual Cyclic Codes Over $M_2(ℤ_4)$. Discussiones Mathematicae. General Algebra and Applications, Tome 42 (2022) no. 2, pp. 349-362. http://geodesic.mathdoc.fr/item/DMGAA_2022_42_2_a7/