On isomorphisms of abelian Cayley objects of certain orders

AbstractLet m be a positive integer such that gcd(m,ϕ(m))=1 (ϕ is Euler's phi function) with m=p1⋯pr the prime power decomposition of m. Let n=p1a1⋯prar. We provide a sufficient condition to reduce the Cayley isomorphism problem for Cayley objects of an abelian group of order n to the prime power case. In the case of Cayley k-ary relational structures (which include digraphs) of abelian groups, this sufficient condition reduces the Cayley isomorphism problem of k-ary relational structures of abelian groups to the prime power case for Cayley k-ary relational structures of abelian groups. As corollaries, we solve the Cayley isomorphism problem for Cayley graphs of Zn (for the specific values of n as above) and prove several abelian groups (for specific choices of the ai) of order n are CI-groups with respect to digraphs

On self-dual affine-invariant codes

AbstractAn extended cyclic code of length 2m over GF(2) cannot be self-dual for even m. For odd m, the Reed-Muller code [2m, 2m−1, 2(m+1)2] is affine-invariant and self-dual, and it is the only such code for m = 3 or 5. We describe the set of binary self-dual affine-invariant codes of length 2m for m = 7 and m = 9. For each odd m, m ⩾ 9, we exhibit a self-dual affine-invariant code of length 2m over GF(2) which is not the self-dual Reed-Muller code. In the first part of the paper, we present the class of self-dual affine-invariant codes of length 2m over GF(2r), and the tools we apply later to the binary codes

Multipliers and generalized multipliers of cyclic objects and cyclic codes

AbstractIn (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 = p2 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 p2 are equivalent by a generalized multiplier times a multiplier. We also develop properties of generalized multipliers and generalized affine maps when n = pm, 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