Cyclic codes over $${\mathbb {F}}_{2^m}[u]/\langle u^k\rangle $$ F 2 m [ u ] / ⟨ u k ⟩ of oddly even length

Abstract

Let F 2 m be a finite field of characteristic 2 andFor any odd positive integer n, it is known that cyclic codes over R of length 2n are identified with ideals of the ring R[x]/ x 2n − 1 . In this paper, an explicit representation for each cyclic code over R of length 2n is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes over R of length 2n is obtained. Moreover, the dual code of each cyclic code and self-dual cyclic codes over R of length 2n are investigated.

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

“…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.…”