SQS-graphs of Solov'eva-Phelps codes

A binary extended 1-perfect code C folds over its kernel via the Steiner quadruple systems associated with its codewords. The resulting folding, proposed as a graph invariant for C, distinguishes among the 361 nonlinear codes C of kernel dimension κ obtained via Solov’eva-Phelps doubling construction, where 9 ≥ κ ≥ 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. 1 Preliminaries, objectives and plan We consider the n-cube Qn as the graph with vertex set F n 2 = {0, 1} n in which each two vertices that differ in exactly one coordinate are joined by an edge. A perfect 1-error-correcting code, or 1-perfect code, C = C of length n = 2 − 1, where 0 < r ∈ Z, is an independent vertex set of Qn such that each vertex of Qn \ C is neighbor of exactly one vertex of C. It follows that C has distance 3 and 2 vertices. Each 1-perfect code C = C of length n = 2 − 1 can be extended by adding an overall parity check. This yields an extended 1-perfect code C = C of length n + 1 = 2, which is a subspace of even-weight words of F 2 . The n + 1 coordinates of the words of F n+1 2 here are orderly indicated 0, 1, . . . , n. For every n = 2 − 1 such that 0 < r ∈ Z there is at least one linear code C as above and a corresponding linear extension, C. These codes are unique for every r < 4. The situation changes for r ≥ 4. In fact, there are many nonlinear codes C and C or length 15 and 16, respectively, [3, 6, 8, 9, 10, 11].