SQS-graphs of Solov'eva-Phelps codes

TL;DR

This paper introduces SQS-graphs derived from Solov'eva-Phelps codes, providing a novel graph invariant that distinguishes among 361 nonlinear codes based on their kernel dimension, using combinatorial and geometric structures.

Contribution

It proposes a new graph invariant for binary extended 1-perfect codes, linking code structure to Steiner quadruple systems and geometric configurations.

Findings

Successfully distinguishes 361 nonlinear codes by kernel dimension

Expresses edges in terms of lexicographically ordered products

Uses geometric structures like Fano plane lines for loops

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 $\kappa$ obtained via Solov'eva-Phelps doubling construction, where $9\geq\kappa\geq 5$. Each of the 361 resulting graphs has most of its nonloop edges expressible in terms of lexicographically ordered quarters of products of classes from extended 1-perfect partitions of length 8 (as classified by Phelps) and loops mostly expressible in terms of the lines of the Fano plane.