On the minimum distance graph of an extended Preparata code

The minimum distance graph of an extended Preparata code P(m) has vertices corresponding to codewords and edges corresponding to pairs of codewords that are distance 6 apart. The clique structure of this graph is investigated and it is established that the minimum distance graphs of two extended Preparata codes are isomorphic if and only if the codes are equivalent.

💡 Research Summary

The paper investigates the minimum‑distance graph (MDG) of the extended Preparata code P(m), where the length is n = 2m, m is even and m ≥ 4. In this graph each vertex corresponds to a codeword of P(m) and two vertices are adjacent precisely when the Hamming distance between the corresponding codewords equals the minimum distance of the code, which for the extended Preparata code is 6. The authors’ main goal is to understand the combinatorial structure of this graph and to determine whether the graph uniquely determines the code up to the usual notion of equivalence (coordinate permutation together with a possible translation).

The paper begins with a concise review of binary vector spaces, Hamming distance, weight, and the definitions of isometry and equivalence for binary codes. It recalls that a 1‑perfect code of length 2m – 1 has minimum distance 3, and its extension (by adding a parity‑check coordinate) has minimum distance 4. The Preparata code P(m) is a nonlinear code of length 2m – 1, minimum distance 5, and its extension P(m) has length 2m and minimum distance 6. It is known from earlier work that any extended Preparata code is a subcode of an extended 1‑perfect code C_P(m). The weight‑4 codewords of C_P(m) form a Steiner quadruple system SQS(2m). Moreover, each block (a 4‑tuple) of the SQS is either not contained in the support of any weight‑6 codeword of P(m) or is contained in exactly one such support.

With these preliminaries, the authors define the MDG DG(P(m)). The all‑zero codeword is taken as a distinguished vertex u₀; its neighbourhood N(u₀) consists of all weight‑6 codewords. Two vertices u, v ∈ N(u₀) are adjacent iff their supports intersect in exactly three coordinates. Consequently, the subgraph induced by N(u₀) is a collection of 6‑vertex cliques whose pairwise intersections are governed by the combinatorics of triples of coordinates.

The core combinatorial analysis is carried out in Section 2. Lemma 2 and Proposition 1 bound the size of any clique inside N(u₀). By examining how many triples can appear in the supports of the vertices of a clique, the authors prove that the maximum possible size of a clique is 13. They then show (Proposition 2) that every maximum‑size clique is precisely the set C(t) of all weight‑6 codewords whose support contains a fixed triple t = {v₁,v₂,v₃}. The size of such a clique is n – 4 choose 3, which for m ≥ 6 exceeds 20, confirming that the bound 13 is tight and that these cliques are indeed maximal.

Having identified the maximal cliques with triples, the paper proceeds to study the interaction between cliques corresponding to different triples. Lemma 3, Proposition 3, 4, 5 and Corollary 3 describe how the number of common coordinates between two triples determines the adjacency pattern between the corresponding cliques. In particular, if two triples share exactly two coordinates, the two cliques intersect in a single vertex and each vertex of one clique has exactly three neighbours in the other; this configuration is shown to be in one‑to‑one correspondence with a block of the Steiner quadruple system. If the triples intersect in at most one coordinate, the neighbour counts drop to at most two, and the cliques are essentially disjoint. These results allow the authors to reconstruct the whole SQS from the MDG.

Section 3 uses the above structural information to “label’’ every vertex of the MDG, i.e., to recover the underlying coordinate set for each codeword. Starting from an arbitrarily chosen maximal clique (say C({1,2,3})), the algorithm labels its vertices as {1,2,3} ∪ t for all 3‑subsets t of the remaining coordinates. Then, by examining the families S({i,j}) of cliques that contain a given pair {i,j}, the algorithm iteratively discovers all other triples and the corresponding cliques, eventually labeling every weight‑6 codeword. Once the weight‑6 layer is labelled, the weight‑4 layer (the SQS blocks) and finally the whole extended Preparata code are recovered.

The final step is the isomorphism theorem: if two extended Preparata codes P(m)₁ and P(m)₂ have isomorphic MDGs, then the above labelling procedure applied to each graph yields identical coordinate assignments up to a permutation of the coordinate set. Consequently, the two codes are equivalent (they differ only by a coordinate permutation and possibly a translation). Conversely, equivalent codes obviously have isomorphic MDGs. Thus the MDG is a complete invariant for the equivalence class of extended Preparata codes.

In summary, the paper makes three significant contributions. First, it identifies the maximal cliques of the MDG as precisely the sets of weight‑6 codewords containing a given triple of coordinates. Second, it establishes a precise correspondence between the interaction of these cliques and the blocks of the Steiner quadruple system associated with the underlying extended 1‑perfect code. Third, it proves that the minimum‑distance graph uniquely determines the extended Preparata code up to equivalence, providing a powerful graph‑theoretic tool for code classification, enumeration, and equivalence testing. This work bridges coding theory and combinatorial design theory, and opens the way for similar analyses of other nonlinear codes whose minimum‑distance graphs may serve as complete invariants.