SQS-graphs of extended 1-perfect codes

TL;DR

This paper introduces SQS-graphs derived from extended 1-perfect codes, providing a new graph invariant that distinguishes among nonlinear codes with specific kernel dimensions, using combinatorial and geometric structures.

Contribution

It proposes a novel folding-based graph invariant for extended 1-perfect codes, linking code structure to Steiner quadruple systems and classical combinatorial designs.

Findings

Successfully distinguishes 361 nonlinear codes by kernel dimension

Expresses edges in terms of lexicographically disjoint quarters of product components

Uses lines of the Fano plane to describe loops in the graphs

Abstract

A binary extended 1-perfect code $\mathcal C$ folds over its kernel via the Steiner quadruple systems associated with its codewords. The resulting folding, proposed as a graph invariant for $\mathcal C$, distinguishes among the 361 nonlinear codes $\mathcal C$ of kernel dimension $\kappa$ with $9\geq\kappa\geq 5$ obtained via Solov'eva-Phelps doubling construction. Each of the 361 resulting graphs has most of its nonloop edges expressible in terms of the lexicographically disjoint quarters of the products of the components of two of the ten 1-perfect partitions of length 8 classified by Phelps, and loops mostly expressible in terms of the lines of the Fano plane.