Coordinate transitivity of extended perfect codes and their SQS

TL;DR

This paper investigates the automorphism groups of extended perfect codes and their associated Steiner quadruple systems, establishing their invariance properties and proposing new constructions for transitive codes.

Contribution

It demonstrates that the automorphism group of the SQS matches that of the code, making SQS a complete invariant, and introduces a new construction method for transitive extended perfect codes.

Findings

Automorphism group of SQS coincides with that of the code.

SQS serve as complete invariants for code isomorphism classes.

New construction for coordinate transitive and neighbor transitive codes.

Abstract

We continue the study of the class of binary extended perfect propelinear codes constructed in the previous paper and consider their permutation automorphism (symmetry) groups and Steiner quadruple systems. We show that the automorphism group of the SQS of any such code coincides with the permutation automorphism group of the code. In particular, the SQS of these codes are complete invariants for the isomorphism classes of these codes. We obtain a criterion for the point transitivity of the automorphism group of SQS of proposed codes in terms of GL-equivalence (similar to EA-type equivalence for permutations of F^r). Based on these results we suggest a new construction for coordinate transitive and neighbor transitive extended perfect codes.