The q-ary image of a q/sup m/-ary cyclic code

Abstract

For (n, q)=1 V a q/sup m/-ary cyclic code of length n and with generator polynomial g(x), we show that there exists a basis for F(q/sup m/) over F/sub q/ with respect to which the q-ary image of V is cyclic, if and only if: (i) g(x) is over F/sub q/; or (ii) g(x)=g/sub 0/(x)(x-/spl gammasup -q(/spl mu/)/), g/sub 0/(x) is over F/sub q/, F/sub qspl ne/F(q/sup k/)=F/sub q/(/spl gamma/)/spl sub/F(q/sup m/), /spl mu/ an integer modulo k, and w/sup m/-/spl gamma/ has a divisor over F(q/sup k/) of degree e=m/k; or (iii) g(x)=g/sub 0/(x) /spl Pisub /spl muspl epsiv/s/(x-/spl gamma/(-q/sup /spl mu)), g/sub 0/(x) is over F/sub q/, F/sub qspl ne/F(q/sup k/)=F/sub q/(/spl gamma/)/spl sub/F(q/sup m/), S a set of integers module k of cardinality k-1 and w/sup m/-/spl mu/ has a divisor over F(q/sup k/) of degree e=m/k. In all of the above cases, we determine all of the bases with respect to which the q-ary image of V is cyclic.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>