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>

Abstract

Let F p m be a finite field of cardinality p m and R = F p m [u]/ u 2 = F p m + uF p m (u 2 = 0), where p is a prime and m is a positive integer. For any λ ∈ F × p m , an explicit representation for all distinct λ-constacyclic codes over R of length np s is given by a canonical form decomposition for each code, where s and n are arbitrary positive integers satisfying gcd(p, n) = 1. For any such code, using its canonical form decomposition the representation for the dual code of the code is provided. Moreover, representations for all distinct cyclic codes, negacyclic codes and their dual codes of length np s over R are obtained, and self-duality for these codes are determined. Finally, all distinct self-dual negacyclic codes over F 5 + uF 5 of length 2 • 3 t • 5 s are listed for any positive integer t.

An explicit representation and enumeration for negacyclic codes of length 2^kn over \mathbb{Z}_4+u\mathbb{Z}_4

“…Proof. A direct proof can be given by an argument paralleling to that of Lemma 3.2, Lemma 3.7 and Theorem 3.8 in [15]. Here, we omitted.…”

“…Then C j is one of the following 113 ideals in Z4[x] fj (−x 2 ) : (j-I) 64 codes: 12 , where β 1 , β 2 ∈ T j . (j-II) 17 codes: 15 , where β 1 ∈ T j . (j-V) 10 codes: 12 , where β 0 ∈ T j .…”

“…The class of constacyclic codes play a very significant role in the theory of errorcorrecting codes. Since 1999, special classes of constacyclic codes over certain classes of finite commutative chain rings have been studied by numerous authors (see [1], [2], [5], [6], [8]- [11], [15], [17]- [19], [21]- [24], for example). It is an important way and an interesting topic to construct optimal codes (over finite fields or finite rings) from special linear codes over some appropriate rings.…”

Constacyclic codes of length $ 8p^s $ over $ \mathbb F_{p^m} + u\mathbb F_{p^m} $

“…In 2018, we established successfully all negacyclic codes of length 4p s over F p m + uF p m [32]. After that, some authors extended these problems to many more general lengths and alphabets (see, e.g., [11], [12], [13], [14], [15], [16], [17], [18], [19], [39], [50]).…”

Bilinear Bäcklund transformation, soliton and breather solutions for a (3+1)-dimensional generalized Kadomtsev-Petviashvili equation in fluid dynamics and plasma physics

“…which models the nonlinear waves in fluid dynamics, plasma physics and weakly dispersive media [10].…”