Explicit Representation and Enumeration of Repeated-Root (δ + αu²)-Constacyclic Codes Over F₂ ^m [u]/‹u ^2λ›

Abstract

Let F 2 m be a finite field of 2 m elements, λ and k be integers satisfying λ, k ≥ 2 and denote R = F 2 m [u]/ u 2λ. Let δ, α ∈ F × 2 m. For any odd positive integer n, we give an explicit representation and enumeration for all distinct (δ + αu 2)-constacyclic codes over R of length 2 k n, and provide a clear formula to count the number of all these codes. In particular, we conclude that every (δ + αu 2)-constacyclic code over R of length 2 k n 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 + u 2)-constacyclic codes of length 4 over F 2 [u] u 4 , and apply our results to determine all 11 self-dual codes among them. INDEX TERMS Type 2 constacyclic code, linear code, repeated-root code, finite chain ring.

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