Multiplier Equivalence Of Cyclic Codes

Abstract

A multiplier is a permutation of S, given by i -+ ai(mod n) where gcd(a,n) = 1. It is known that if two cyclic codes of prime length p are equivalent, then they must also be multiplier equivalent. We show that this same result holds for cyclic codes of length n whenever (n,d(n)) = 1. Let C and C’ be cyclic codes of length pr with p and T primes, p > T, so that the Sylow p-subgroup of the group of C has order p. Then if C and C‘ are equivalent they must be equivalent by a multiplier. We introduce generalized multipliers for cyclic codes of length pm where p is prime and demonstrate many of their properties. In particular, we show that, under natural conditions, generalized multipliers map cyclic codes onto cyclic codes and that certain generalized multipliers are in the group of any cyclic code. In the special case m = 2 we show that two equivalent cyclic codes of length p2 are equivalent by a multiplier or a generalized multiplier times a multiplier. Examples are given of cyclic codes of length p2 which are equivalent by a generalized multiplier but not by a multiplier. By examining a matrix equation we find a class of binary cyclic codes which are multiplier equivalent whenever they are equivalent.