SQS-graphs of Solov'eva-Phelps codes

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 $9\ge \kappa \ge 5$. 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.

14 pages, 15 tables