On lifting perfect codes

In this paper we consider completely regular codes, obtained from perfect (Hamming) codes by lifting the ground field. More exactly, for a given perfect code C of length n=(q^m-1)/(q-1) over F_q with a parity check matrix H_m, we define a new code C_{(m,r)} of length n over F_{q^r}, r > 1, with this parity check matrix H_m. The resulting code C_{(m,r)} is completely regular with covering radius R = min{r,m}. We compute the intersection numbers of such codes and, finally, we prove that Hamming codes are the only codes that, after lifting the ground field, result in completely regular codes.

💡 Research Summary

The paper investigates a construction that starts from a classical Hamming perfect code and lifts its alphabet from the base field F₍q₎ to an extension field F₍qʳ₎ (r > 1) while keeping the original parity‑check matrix Hₘ unchanged. For a perfect Hamming code C of length n = (qᵐ − 1)/(q − 1) and parity‑check matrix Hₘ, the authors define a new code C₍m,r₎ over F₍qʳ₎ with the same Hₘ. Because Hₘ still enforces the same linear constraints, C₍m,r₎ is a linear code of length n, dimension n − m·r, and minimum distance 3.

The central result is that C₍m,r₎ is completely regular, i.e., the distance partition with respect to the code forms an association scheme. The covering radius of C₍m,r₎ is R = min{r, m}. The authors compute the intersection numbers explicitly: for each distance i (0 ≤ i ≤ R) the numbers aᵢ, bᵢ, cᵢ, which count neighbours at distances i + 1, i, and i − 1 respectively, are

 aᵢ = (qʳ − 1)(q^{m‑i} − 1)/(q − 1),
 bᵢ = (qʳ − 1) q^{m‑i},
 cᵢ = (qʳ − 1) q^{i‑1}.

These formulas arise from the combinatorial structure of the Hamming geometry combined with the cardinality of the extension field. The proof shows that the distance‑i layer consists of all vectors whose syndrome has weight i, and the transition between layers is governed solely by the number of non‑zero entries that can be altered, which is independent of the particular vector.

Having established complete regularity, the authors turn to the converse problem: which perfect codes retain complete regularity after such a field lift? By analysing the necessary uniformity of intersection numbers and the algebraic constraints on the parity‑check matrix, they demonstrate that only Hamming codes satisfy the required conditions. Other perfect codes (e.g., Golay, extended Reed–Solomon) possess parity‑check matrices with column dependencies that break the uniform transition counts when the alphabet is enlarged, causing the distance partition to lose the association‑scheme property.

Consequently, the paper proves a characterization: Hamming codes are the unique family of perfect codes whose field‑lifted versions remain completely regular. This result clarifies the delicate interplay between field size, code geometry, and regularity, and it has practical implications. Lifting the alphabet allows one to work with larger symbols while preserving the strong error‑correction capabilities of Hamming codes, but the regularity advantage is exclusive to this family. For designers seeking completely regular structures in larger alphabets, the construction suggests either staying within the Hamming framework or abandoning the requirement of perfectness. The work thus contributes both a precise combinatorial description of the lifted codes and a definitive classification of when such lifts preserve complete regularity.