Constacyclic codes of length ps over Fpm+uF

Abstract

For any prime p, all constacyclic codes of length p s over the ring R = F p m + uF p m are considered. The units of the ring R are of the forms γ and α + uβ, where α, β, and γ are nonzero elements of F p m , which provides p m (p m − 1) such constacyclic codes. First, the structure and Hamming distances of all constacyclic codes of length p s over the finite field F p m are obtained; they are used as a tool to establish the structure and Hamming distances of all (α + uβ)-constacyclic codes of length p s over R. We then classify all cyclic codes of length p s over R and obtain the number of codewords in each of those cyclic codes. Finally, a one-to-one correspondence between cyclic and γ -constacyclic codes of length p s over R is constructed via ring isomorphism, which carries over the results regarding cyclic codes corresponding to γ -constacyclic codes of length p s over R.

Recent progress on weight distributions of cyclic codes over finite fields

“…In particular, for Later on, Dinh in [30] showed that all constacyclic codes of length p s over F q are scalar equivalent to negacyclic codes. , given by Φ(f (X)) = f (λ 0 X).…”

Skew Constacyclic Codes over Finite Fields and Finite Chain Rings

“…In the next part, the structure of skew Θ-cyclic and negacyclic codes over F + F is studied. We refer the readers to [26,38] We categorize the left ideals of (F + F )[ ; Θ]/⟨ − ⟩ into three types: Type LI-1 refers to the trivial ideal (⟨0⟩, ⟨1⟩) or a left ideal satisfying part (i) of the theorem above. Similarly, LI-2 and LI-3 refer to a left ideal satisfying Theorem 52 ((ii) and (iii)), respectively.…”

Constacyclic codes of length <inline-formula><tex-math id="M1">\begin{document}$np^s$\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M2">\begin{document}$\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$\end{document}</tex-math></inline-formula>

“…We will identify a with a(x) in this paper. By [9] In this paper, let F p m be a finite field of cardinality p m , where p is a prime and m is a positive integer, and denote F p m [u]/ u 2 by F p m + uF p m (u 2 = 0). There were a lot of literatures on linear codes, cyclic codes and constacyclic codes of length N over rings F p m + uF p m (u 2 = 0) for various prime p and positive integers m and some positive integer N .…”

“…For example, all constacyclic codes of length 2 s over the Galois extension rings of F 2 + uF 2 were classified and their detailed structures were also established in [8]. Dinh [9] classified all constacyclic codes of length p s over F p m + uF p m . Dinh et al [10] studied negacyclic codes of length 2p s over the ring F p m + uF p m .…”

“…Recently, Dinh et al [11] [12] studied constacyclic codes of length 4p s over F p m + uF p m . These papers mainly used the methods in [8] and [9], and the main results and their proofs depend heavily on the code lengths p s , 2p s and 4p s .…”