A Hyperplane Bound and Ward's Divisibility Conjecture
Abstract
Let q=pf, where p is prime. We prove that every q-ary linear code meeting the Griesmer bound, with positive dimension and minimum distance d≥1, is pmax{0,νp(d)−f+1}-divisible, as asserted in Ward's conjecture. Arbitrary projective column multiplicities are allowed. The geometric ingredient is a bound independent of dimension: a nonzero scalar-invariant function F:𝔽qk→𝔽p, with F(0)=0 and prime-field algebraic degree at most (f+s)(p−1), vanishes identically on at most ps(q−1) linear hyperplanes when 0≤ s<f. A truncated digit-sum inequality forces selected coefficient polynomials to be homogeneous of ordinary degree less than q. Weighted moments on a projective line then bound their linear factors, while a second-moment argument handles the complementary support configuration. An elementary finite-difference calculation supplies the degree bound for divided weight residues. Endpoint shortenings prove the basic divisibility layers; endpoint lifting, a surjective linear functional, and residual-code induction extend the argument to every valuation. We include the proofs of the classical ingredients, identify their sources, and distinguish the present argument from an earlier reported unpublished proof.
Related
- On the proof of multi-variable n/d-conjecture for hyperplane arrangements
- Suciu's Conjecture on hyperplane arrangements
- The strong monodromy conjecture for hyperplane arrangements
- A proof of Ward's conjecture on Griesmer codes
- Multivariate V-filtrations and the Strong Monodromy Conjecture for hyperplane arrangements
BibTeX
@misc{zhang2026hyperplane,
title = {{A Hyperplane Bound and Ward's Divisibility Conjecture}},
author = {Guanghui Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26200},
eprint = {2610.26200},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26200},
}
RIS
TY - UNPB
TI - A Hyperplane Bound and Ward's Divisibility Conjecture
AU - Guanghui Zhang
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26200
UR - https://arxiv.org/abs/2610.26200
AB - Let q=p^f, where p is prime. We prove that every q-ary linear code meeting the Griesmer bound, with positive dimension and minimum distance d≥1, is p^max{0,ν_p(d)−f+1}-divisible, as asserted in Ward's conjecture. Arbitrary projective column multiplicities are allowed. The geometric ingredient is a bound independent of dimension: a nonzero scalar-invariant function F:𝔽_q^k→𝔽_p, with F(0)=0 and prime-field algebraic degree at most (f+s)(p−1), vanishes identically on at most p^s(q−1) linear hyperplanes when 0≤ s<f. A truncated digit-sum inequality forces selected coefficient polynomials to be homogeneous of ordinary degree less than q. Weighted moments on a projective line then bound their linear factors, while a second-moment argument handles the complementary support configuration. An elementary finite-difference calculation supplies the degree bound for divided weight residues. Endpoint shortenings prove the basic divisibility layers; endpoint lifting, a surjective linear functional, and residual-code induction extend the argument to every valuation. We include the proofs of the classical ingredients, identify their sources, and distinguish the present argument from an earlier reported unpublished proof.
ER -