A mathematical preprint reports a proof of a cited conjecture on the minimum distance of certain evaluation codes. The authors say the result comes from a refined generalized Bézout bound—a way of limiting how many common projective zeros an overdetermined polynomial system can have—and that the same approach also yields lower bounds for generalized Hamming weights.
Counting the points
The mathematical setting is evaluation of homogeneous forms at points of zero-dimensional reduced complete intersections defined over the finite field Fq. The projective evaluation code is built by evaluating those forms at chosen representatives of the projective points; changing the representatives only produces a monomially equivalent code, preserving its dimension and generalized Hamming weights.
At the algebraic core, the paper establishes a cardinality bound for the common-zero set of an overdetermined system. In plain terms, the number of projective common zeros is limited by a product of the equations’ degrees; when an initial group of equations forms a regular sequence, it gives a corresponding product bound using those degrees and the remaining degree factors.
From geometry to code weights
The paper then considers codes evaluating degree-d homogeneous forms on zero-dimensional reduced complete intersections over Fq. Under the ordering 2 ≤ d₁ ≤ d₂ ≤ … ≤ dₖ₋₁ for the relevant defining degrees, it obtains a lower bound for the code’s minimum distance, the quantity at the heart of the cited conjecture.
In the paper’s notation, the bound is d₁(C) ≥ (d₁ − min{d,d₁})d₂⋯dₖ₋₁. The authors state that this proves the cited minimum-distance conjecture, while the result is presented as a lower bound rather than an equality in every qualifying case.
The paper also tracks generalized Hamming weights through a characterization based on how many columns of a generator matrix can lie in a vector space of dimension k−r. For evaluation codes built from homogeneous linear forms, it gives, for 1 ≤ r ≤ k−1, the bound dᵣ(C) ≥ (d₁⋯dᵣ − 1)dᵣ₊₁⋯dₖ₋₁.
For a reduced point set, the code is nondegenerate and the top generalized Hamming weight equals the code length: dₖ(C) = n = d₁⋯dₖ₋₁. The generalized-weight result does not cover arbitrary-degree form evaluations; the paper says the same proof route cannot simply be extended because linearly independent homogeneous forms of the same degree need not form a regular sequence.
What remains unsettled
Those results do not settle the paper’s broader conjecture. It proposes that projective Cartesian codes have the smallest generalized Hamming weights among evaluation codes supported on reduced complete intersections with the same degrees.
The manuscript says the conjecture’s minimum-distance part holds in selected cases, including when X is contained in P², when d<d₁ for any k, and when d=1. The full projective-Cartesian minimality statement remains a conjecture in the paper.
The results are theoretical bounds about mathematical objects, not measurements of decoding, noisy-channel performance or deployment. They also do not establish that the lower bounds are tight for every qualifying complete intersection.
The document is identified as arXiv:2608.19978v1 and dated 20 Aug 2026.
Paper data and sources
Original title: Generalized Hamming weights of codes arising from complete intersection
Authors: Eduardo Camps Moreno, Flavio Salizzoni, Rodrigo San-José
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text