SQS-graphs of extended 1-perfect codes

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.

💡 Research Summary

The paper introduces a novel graph‑theoretic invariant for binary extended 1‑perfect codes, called the SQS‑graph, and demonstrates its power in distinguishing a large family of nonlinear codes obtained via the Solov’eva‑Phelps doubling construction. An extended 1‑perfect code 𝒞 is a binary code of length 2ⁿ that attains the perfect packing bound after an overall parity‑check bit is added. The kernel Ker(𝒞) (the set of codewords that remain unchanged under component‑wise addition with any codeword) plays a central role: the authors “fold” the code over its kernel, i.e., they consider the quotient 𝒞/Ker(𝒞) and study the interaction of the resulting cosets through the Steiner quadruple systems (SQS) naturally associated with each codeword.

For each codeword c∈𝒞, the support of c determines a collection of 4‑element subsets that form a Steiner quadruple system S(c). Two cosets A and B of the kernel are joined by a non‑loop edge in the SQS‑graph G(𝒞) precisely when the SQS of any representative of A shares at least one quadruple with the SQS of any representative of B. A loop at a vertex A appears when the SQS of A contains a line of the Fano plane (the unique 7‑point projective plane of order 2). Thus the vertex set of G(𝒞) is the set of kernel cosets, edges encode shared quadruples, and loops encode Fano‑plane lines.

The authors then give an explicit combinatorial description of the edges and loops for the 361 nonlinear codes whose kernel dimension κ satisfies 5≤κ≤9. These codes arise from the Solov’eva‑Phelps construction, which doubles a 1‑perfect code of length 2ⁿ⁻¹ using a pair of 1‑perfect partitions of length 8. Phelps previously classified the ten inequivalent 1‑perfect partitions of length 8 (denoted 𝔓₁,…,𝔓₁₀). By taking the Cartesian product 𝔓ᵢ×𝔓ⱼ, one obtains 64 ordered pairs of 4‑element blocks. The authors focus on “lexicographically disjoint quarters” of these products: each product can be split into four quarters according to the binary ordering of the coordinates, and the quarters that are disjoint in the lexicographic sense contain the quadruples that generate the majority of non‑loop edges in G(𝒞). In other words, most edges are accounted for by the intersection patterns of blocks that lie in a specific quarter of a product of two partitions.

Loops are treated more uniformly. The Fano plane has seven lines, each line being a 3‑point set. When a coset’s SQS contains any of these three‑point configurations (viewed as the projection of a quadruple onto three coordinates), a loop is placed at that vertex. Consequently, the loop structure of G(𝒞) is almost entirely determined by the incidence structure of the Fano plane.

The paper provides exhaustive computational data: for each of the 361 codes the authors construct G(𝒞), list its edge multiplicities, and verify that the resulting graphs are pairwise non‑isomorphic. This demonstrates that the SQS‑graph is a strictly finer invariant than the traditional parameters (length, dimension, minimum distance, kernel size). In several cases, two codes share all classical invariants yet yield distinct SQS‑graphs, confirming the discriminating power of the new invariant.

Beyond classification, the authors analyze the automorphism groups of the SQS‑graphs. They show that the automorphism group of G(𝒞) is generated by the symmetries of the underlying 1‑perfect partitions and the automorphisms of the Fano plane. This reveals that the folding process preserves the symmetries of the original code: any automorphism of 𝒞 that fixes the kernel descends to an automorphism of G(𝒞). Conversely, extra symmetries of G(𝒞) can sometimes be lifted to new automorphisms of the code, offering a pathway to construct codes with prescribed symmetry properties.

Finally, the authors discuss potential extensions. The SQS‑graph construction relies only on the existence of a Steiner system associated with each codeword and on a nontrivial kernel. Therefore, it can be adapted to other families of perfect or near‑perfect codes, to nonbinary alphabets, and to codes with larger kernels. They conjecture that for any perfect code with a nontrivial kernel, the corresponding SQS‑graph will capture subtle combinatorial information that is invisible to standard algebraic invariants.

In summary, the paper makes three major contributions: (1) it defines the SQS‑graph as a new, graph‑based invariant for extended 1‑perfect codes; (2) it provides a detailed combinatorial description of the edges and loops of these graphs using the ten length‑8 1‑perfect partitions and the Fano plane; and (3) it demonstrates, through exhaustive computation, that the SQS‑graph distinguishes all 361 nonlinear extended 1‑perfect codes with kernel dimensions 5 through 9 obtained by the Solov’eva‑Phelps construction. This work bridges coding theory, design theory, and graph theory, and opens new avenues for the classification and analysis of highly structured error‑correcting codes.