Designs and codes from fixed points of alternating groups

Abstract

In this paper, we construct some designs invariant under the alternating groups, using a method that has been recently introduced by Moori. We consider the action of A_n on the 2-subsets of a set of size n. The blocks of the designs are the conjugates of the set of fixed-points of the permutation with respect to this action. For every element $g\in A_n,$ we give explicit formulae to compute the parameters of the designs based on the cycle representation of g. We also prove that the binary codes of these designs have minimum distances at most 4.

Keywords:

• Alternating group
• code
• design
• fixed point

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 05B05
• 20D06

Acknowledgments

The author is very grateful to Professor J. Moori for his helpful suggestions. He also thanks the referee for the constructive comments.