Cyclic codes of length 5p s over F p m + uF p m and their duals

Hai Q Dinh ; Bac T Nguyen ; Roengchai Tansuchat ; Hiep L Thi

Filomat, Tome 37 (2023) no. 26, p. 9009

Résumé

For an odd prime p 5, the structures of cyclic codes of length 5p s over R = F p m + uF p m (u 2 = 0) are completely determined. Cyclic codes of length 5p s over R are considered in 3 cases, namely, p ≡ 1 (mod 5), p ≡ 4 (mod 5), p ≡ 2 or 3 (mod 5). When p ≡ 1 (mod 5), a cyclic code of length 5p s over R can be expressed as a direct sum of a cyclic code and γ p s i-constacyclic codes of length p s over R, where γ p s i = − i(p m −1)p s 10 , i = 1, 3, 7, 9. When p ≡ 4 (mod 5), it is equivalent to p m ≡ 1 (mod 5) when m is even and p m ≡ 4 (mod 5) when m is odd. If p m ≡ 1 (mod 5) when m is even, then a cyclic code of length 5p s over R can be obtained as a direct sum of a cyclic code and γ p s i-constacyclic codes of length p s over R, where γ p s i = − i(p m −1)p s 10 , i = 1, 3, 7, 9. If p m ≡ 4 (mod 5) when m is odd, then a cyclic code of length 5p s over R can be expressed as a direct sum of a cyclic code of length p s over R and an α 1 and α 2-constacyclic code of length 2p s over R, for some α 1 , α 2 ∈ F p m {0}. If p ≡ 2 or 3 (mod 5) such that p m 1 (mod 5), then a cyclic code of length 5p s over R can be expressed as C 1 ⊕ C 2 , where C 1 is an ideal of R[x] 〈x p s −1〉 and C 2 is an ideal of R[x] 〈(x 4 +x 3 +x 2 +x+1) p s 〉. We also investigate all ideals of R[x] 〈(x 4 +x 3 +x 2 +x+1) p s 〉 to study detail structure of a cyclic code of length 5p s over R. In addition, dual codes of all cyclic codes of length 5p s over R are also given. Furthermore, we give the number of codewords in each of those cyclic codes of length 5p s over R. As cyclic and negacyclic codes of length 5p s over R are in a one-by-one equivalent via the ring isomorphism x → −x, all our results for cyclic codes hold true accordingly to negacyclic codes.

Détail

DOI : 10.2298/FIL2326009D

Classification : 94B15, 94B05, 11T71
Keywords: Cyclic codes, constacyclic codes, dual codes, repeated-root codes