On symmetry group of Mollard code

TL;DR

This paper investigates the symmetry groups of Mollard codes, generalizing previous results on Vasilev codes and analyzing the linearity of coordinate positions in nonlinear perfect codes.

Contribution

It extends the understanding of symmetry groups of Mollard codes and explores the linearity of coordinate positions in nonlinear perfect codes.

Findings

Determined the group Stab_{D2}Sym(M(C,D)) for Mollard codes.

Generalized symmetry group results from Vasilev codes.

Discussed linearity of coordinate positions in nonlinear perfect codes.

Abstract

For a pair of given binary perfect codes C and D of lengths t and m respectively, the Mollard construction outputs a perfect code M(C,D) of length tm + t + m, having subcodes C1 and D2, that are obtained from codewords of C and D respectively by adding appropriate number of zeros. In this work we generalize of a result for symmetry groups of Vasilev codes [2] and find the group Stab_{D2}Sym(M(C,D)). The result is preceded by and partially based on a discussion of linearity of coordinate positions (points) in a nonlinear perfect code (non-projective Steiner triple system respectively).