A Hyperplane Bound and Ward's Divisibility Conjecture

Abstract

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 $d\ge1$, is $p^{\max\{0,\nu_p(d)-f+1\}}$-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 $F:\mathbb{F}_q^k\to\mathbb{F}_p$, 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 $0\le s<f$. 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of the classical ingredients, identify their sources, and distinguish the present argument from an earlier reported unpublished proof.