Explicit representation and enumeration of repeated-root (δ + αu²)-Constacyclic Codes over F<inf>2</inf>^m[u]/⟨u^2?⟩

Abstract

© 2013 IEEE. Let F2m be a finite field of 2m elements, λ and k be integers satisfying λ,k ≥ 2 and denote R= F2m[u]/&lsaquo; u2λ &rsaquo;. Let δ,α F2m×. For any odd positive integer n, we give an explicit representation and enumeration for all distinct (δ +α u2)-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 +α u2)-constacyclic code over R of length 2kn is an ideal generated by at most 2 polynomials in the ring R[x]/&lsaquo; x2kn-(δ +α u2)&rsaquo;. As an example, we listed all 135 distinct (1+u2)-constacyclic codes of length 4 over F2[u]/&lsaquo; u4&rsaquo;, and apply our results to determine all 11 self-dual codes among them.