Explicit Representatives and Sizes of Cyclotomic Cosets II: Cyclotomic Systems and Applications to Constacyclic Codes

Abstract

Let $q=p^e$ be a prime power, let $\ell$ be a prime with $\gcd (\ell ,qn)=1$, and let $n$ be a positive integer. While $q$-cyclotomic cosets are usually considered modulo a fixed integer, we consider their behavior along the sequence $n,\ell n,{\ell }^{2}n,\dots$ of moduli. To describe the resulting compatibility among cyclotomic cosets, we introduce the $\ell$-adic $q$-cyclotomic system with base module $n$, defined as the projective limit of the spaces of $q$-cyclotomic cosets modulo ${\ell }^{i}n$. We associate to each $q$-cyclotomic coset modulo $n$ a cyclotomic $\ell$-adic integer and use its $\ell$-adic expansion to describe all compatible liftings through the successive levels. This leads to a classification of the corresponding sequences according to their splitting and stable behaviors, together with explicit formulas for the representatives and sizes of their components. We treat separately the cases of odd $\ell$ and $\ell =2$. These results are then used to determine representatives and sizes of $q$-cyclotomic cosets for arbitrary admissible parameters. As an application, we obtain explicit irreducible factorizations of binomials over finite fields and use them to describe the corresponding constacyclic codes.