Reconstructing Extended Perfect Binary One-Error-Correcting Codes from Their Minimum Distance Graphs

The minimum distance graph of a code has the codewords as vertices and edges exactly when the Hamming distance between two codewords equals the minimum distance of the code. A constructive proof for reconstructibility of an extended perfect binary one-error-correcting code from its minimum distance graph is presented. Consequently, inequivalent such codes have nonisomorphic minimum distance graphs. Moreover, it is shown that the automorphism group of a minimum distance graph is isomorphic to that of the corresponding code.

💡 Research Summary

The paper investigates the relationship between extended perfect binary one‑error‑correcting codes and their minimum‑distance graphs (MDGs). After introducing basic notions—binary codes, Hamming distance, MDGs, and Steiner systems—the authors reformulate the classic equivalence problem in terms of weak isometries (preserving only the minimum distance) and full isometries (preserving all distances). Building on earlier work that showed weakly isometric perfect codes are actually isometric, the authors first prove that for extended 1‑perfect codes the Hamming distance between any two codewords can be deduced solely from the MDG. By counting neighbors at distances i‑2, i, i+2, and i+4, they establish a strict inequality that separates the cases i+2 and i+4 for i ≥ 4, yielding Theorem 1 (weakly isometric ⇒ isometric) and, via known results, Theorem 2 (weakly isometric ⇒ equivalent for n ≥ 256).

The core contribution is a constructive reconstruction algorithm (Theorem 3). Starting from an arbitrary vertex identified as the all‑zero word, Lemma 1 allows detection of codewords at distance 6, while Lemma 2 and Lemma 3 show that the block graph of the neighbourhood Steiner quadruple system (SQS) can be recovered from the MDG and that maximal cliques in this block graph uniquely correspond to points of the SQS for v ≥ 16. Consequently, all codewords of weight ≤ 4 are obtained, and higher‑weight words are reconstructed inductively by examining triples of coordinates that fail to form a block in the neighbourhood SQS. This process yields a full reconstruction of the extended code up to equivalence.

The paper then reduces the reconstruction of ordinary 1‑perfect codes to the extended case. Lemma 4 demonstrates that pairs of codewords at Hamming distance 4 can be identified in the MDG of a 1‑perfect code; by adding edges for these pairs one obtains the MDG of the extended code (obtained by appending a parity coordinate). Applying Theorem 3 to this augmented graph reconstructs the extended code, after which the parity coordinate is removed, giving Theorem 4 (reconstruction of 1‑perfect codes).

Finally, the authors address automorphism groups. They prove that the automorphism group of an extended 1‑perfect code (n ≥ 16) is isomorphic to the automorphism group of its MDG (Theorem 5), and similarly for 1‑perfect codes with n ≥ 15 (Theorem 6). The proofs rely on the explicit reconstruction: any graph automorphism induces a code automorphism and vice versa.

In conclusion, the paper shows that the MDG contains complete information about (extended) perfect binary 1‑error‑correcting codes: it determines the code up to equivalence, distinguishes inequivalent codes via non‑isomorphic MDGs, and captures the full symmetry group. This reduces the traditionally difficult code equivalence problem to the well‑studied graph isomorphism problem, offering both theoretical insight into metric rigidity and practical benefits for code classification.