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Explicit Representation and Enumeration of Repeated-Root (δ + αu²)-Constacyclic Codes Over F₂m[u]/‹u

Authors: 

Yuan Cao, Yonglin Cao, Hai Q. Dinh, Tushar Bag, Woraphon Yamaka

Source title: 
IEEE Access, 8: 55550-55562, 2020 (ISI)
Academic year of acceptance: 
2020-2021
Abstract: 

Let F2(m) be a finite field of 2m elements, λ and k be integers satisfying λ, k ≥ 2 and denote R = F2(m) [u]/(u ). Let δ, α ∈ F2(m)× . For any odd positive integer n, we give an explicit representation and enumeration for all distinct (δ +αu2)-constacyclic codes over R of length 2kn, and provide a clear formula to count the number of all these codes. In particular, we conclude that every (δ + αu2)-constacyclic code over R of length 2kn is an ideal generated by at most 2 polynomials in the ring R[x]/〈x 2(k)n - (δ + αu 2 )〉. As an example, we listed all 135 distinct (1 + u2)-constacyclic codes of length 4 over F2[u]/〈u 4 〉, and apply our results to determine all 11 self-dual codes among them.