Asymptotically Sharp Length Thresholds for \texorpdfstring{q}{q}-ary Codes with Distances \texorpdfstring{$22 and texorpdfstring4$}{4}
Abstract
For an integer q≥5, let Aq(n,{2,4}) be the largest cardinality of a subset of a q-ary Hamming space of length n in which every two distinct words have distance $2$ or $4$. Set N(q)=⌈max{2q+1, (17+√(8q2+96q−95))/2}⌉. We prove that Aq(n,{2,4})=1+binom n2 for every n≥ N(q). In particular, N(q)=2q+1 for q≥26. An explicit code of size q(n−2) shows that the least eventual threshold N*(q) satisfies N*(q)≥2q−1; consequently 2q−1≤ N*(q)≤2q+1 for every q≥26. The argument applies to arbitrary integer alphabet sizes, without a linearity assumption. Its main ingredients are synchronization of symbols on intersecting supports, weighted bounds for $2$-intersecting families of four-sets, and a joint estimate for the weight-two and weight-four layers in a common-three-core configuration. No computational verification is required.
Related
- Cislunar navigation is an observability problem: arc-length thresholds, Gramian conditioning, and the Doppler lever
- Radial Convergence along Decreasing Coordinate Radii in \texorpdfstring{H∞(T∞)}{H-infinity(T-infinity)}
- The Covering Radius of the Reed--Muller Code \texorpdfstring{RM(3,8)}{RM(3,8)} is 56
- Transfinite contractive propagation and the ball fixed point property in \texorpdfstring{C(K)}{C(K)}
- Pinned distances of planar sets with low dimension
BibTeX
@misc{djaljapayan2026asymptotically,
title = {{Asymptotically Sharp Length Thresholds for \texorpdfstring$q$q-ary Codes with Distances \texorpdfstring$2$2 and \texorpdfstring$4$4}},
author = {Valjakas Djaljapayan and Yen-Chi Roger Lin and Wei-Hsuan Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30613},
eprint = {2610.30613},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30613},
}
RIS
TY - UNPB
TI - Asymptotically Sharp Length Thresholds for \texorpdfstring{q}{q}-ary Codes with Distances \texorpdfstring{$22 and texorpdfstring4$}{4}
AU - Valjakas Djaljapayan
AU - Yen-Chi Roger Lin
AU - Wei-Hsuan Yu
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30613
UR - https://arxiv.org/abs/2610.30613
AB - For an integer q≥5, let A_q(n,{2,4}) be the largest cardinality of a subset of a q-ary Hamming space of length n in which every two distinct words have distance $2$ or $4$. Set N(q)=⌈max{2q+1, (17+√(8q^2+96q−95))/2}⌉. We prove that A_q(n,{2,4})=1+binom n2 for every n≥ N(q). In particular, N(q)=2q+1 for q≥26. An explicit code of size q(n−2) shows that the least eventual threshold N_*(q) satisfies N_*(q)≥2q−1; consequently 2q−1≤ N_*(q)≤2q+1 for every q≥26. The argument applies to arbitrary integer alphabet sizes, without a linearity assumption. Its main ingredients are synchronization of symbols on intersecting supports, weighted bounds for $2$-intersecting families of four-sets, and a joint estimate for the weight-two and weight-four layers in a common-three-core configuration. No computational verification is required.
ER -