The Ternary and Quaternary Cases of the {2,4}-Distance Code Conjecture
Abstract
For an integer q≥2, let Aq(n,{2,4}) be the largest cardinality of an unrestricted q-ary code of length n in which any two distinct words have Hamming distance $2$ or $4$. We prove that Aq(n,{2,4})=1+binom n2 (q∈{3,4}, n≥6), thereby settling the two remaining alphabet sizes in the conjecture of Boyvalenkov, Delchev, Zinoviev, and Zinoviev; the binary case was settled previously. The quaternary result implies the ternary result by alphabet inclusion. For n≥16, the quaternary upper bound is proved combinatorially. After normalizing one word to zero, the weight-four supports form a $2$-intersecting family; exact support intersections force symbol synchronization and restrict support multiplicities. We combine the nontrivial intersection theorem with matching arguments and a three-coordinate projection lemma. The finite lengths 6≤ n≤13 are handled by a complete Hamming-isometry reduction to exact clique decisions, and n=14,15 by a smaller exact weighted-clique relaxation. All finite decisions are reproduced by two standalone integer-only C++17 verifiers provided as supplementary source files.
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BibTeX
@misc{lin2026ternary,
title = {{The Ternary and Quaternary Cases of the $\2,4\$-Distance Code Conjecture}},
author = {Yen-Chi Roger Lin and Valjakas Djaljapayan and Wei-Hsuan Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30615},
eprint = {2610.30615},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30615},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Ternary and Quaternary Cases of the {2,4}-Distance Code Conjecture
AU - Yen-Chi Roger Lin
AU - Valjakas Djaljapayan
AU - Wei-Hsuan Yu
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30615
UR - https://arxiv.org/abs/2610.30615
AB - For an integer q≥2, let A_q(n,{2,4}) be the largest cardinality of an unrestricted q-ary code of length n in which any two distinct words have Hamming distance $2$ or $4$. We prove that A_q(n,{2,4})=1+binom n2 (q∈{3,4}, n≥6), thereby settling the two remaining alphabet sizes in the conjecture of Boyvalenkov, Delchev, Zinoviev, and Zinoviev; the binary case was settled previously. The quaternary result implies the ternary result by alphabet inclusion. For n≥16, the quaternary upper bound is proved combinatorially. After normalizing one word to zero, the weight-four supports form a $2$-intersecting family; exact support intersections force symbol synchronization and restrict support multiplicities. We combine the nontrivial intersection theorem with matching arguments and a three-coordinate projection lemma. The finite lengths 6≤ n≤13 are handled by a complete Hamming-isometry reduction to exact clique decisions, and n=14,15 by a smaller exact weighted-clique relaxation. All finite decisions are reproduced by two standalone integer-only C++17 verifiers provided as supplementary source files.
ER -