Multipliers and generalized multipliers of cyclic objects and cyclic codes

Abstract

In ( European J. Combin. Theory 8 (1987), 35–43) Pálfy answers the question: Under what conditions on n is it true that two equivalent objects in any class of cyclic combinatorial objects on n elements implies the objects are equivalent, using one of the ϕ( n) multipliers i → ai mod, n, with gcd( a, n) = 1. Pálfy proved that this is true precisely when n = 4 or gcd( n, ϕ( n)) = 1. For any odd prime p, we prove that two equivalent objects in any class of cyclic combinatorial objects on n = p 2 elements are equivalent using a permutation from a list of no more then ϕ( n) = p( p − 1) permutations. We introduce permutations called generalized multipliers, and we show that two permutation equivalent cyclic codes of length p 2 are equivalent by a generalized multiplier times a multiplier. We also develop properties of generalized multipliers and generalized affine maps when n = p m , show that they map cyclic codes to cyclic codes, and show that certain of these maps are in the automorphism group of a cyclic code.