Hulls, linear equivalence, and weighted superelliptic codes
Abstract
The containment of the code of the meet G∧ A in the hull and the identity G∨ A−D=K−G∧ A exchanging meet and join are known; imposing that G∧ A be principal constructs algebraic geometry codes with one-dimensional hull. We turn that construction into a measurement. For arbitrary divisors G and A we compute CL(D,G)∩ CL(D,A) exactly: it is the code of the meet together with an excess ε(G,A), canonically their quotient and a subquotient of H1(𝒪(G∧ A)). So ε vanishes exactly when the meet is non-special; otherwise it certifies that K−G∧ A is linearly equivalent to an effective divisor, at degree zero the vanishing of a single class in the Picard group: the hull detects a linear equivalence rather than being built from one. For superelliptic curves yn=f(x) both sides can be computed: their weighted plane models in ℙ2(1,n/c,d/c), c=gcd(n,d), identify codes Cs of weighted forms of degree s with those of sD∞ and turn hulls into lattice counts. The range on which ε is blind is an explicit interval of degrees, where dimHull(Cs)=cμ(s)−nδ+1−gX, μ(s)=min{s,M−s}, depends only on its affine-point count. Outside it the meet and join are invariant under s↦ M−s while ε is not, so every asymmetry of the hull profile is excess and the threshold in s refines the divisor class: two totally split curves of genus two, over 𝔽7 and over 𝔽11, present the same class at the same pair of degrees and are separated by the profile alone. If 0≤°(G∧ A)≤2gX−2 the hull is at most gX+1, so it is large only where it is blind, and over a prime field, under an explicit inequality on (n,d,q), its maximum over the family is ℓ(⌊ M/2⌋ D∞), attained exactly on the totally split locus.
Related
- Code Equivalence and Automorphism Groups of Weighted Superelliptic Codes
- Hermitian Hulls of Cyclic Codes and Exact Dual Distances of Extended Linear Codes
- Linear Equivalence of Full Weight Spectrum Cyclic Orbit Codes
- Local unitary equivalence of orthogonal arrays and related linear codes
- Galois Hulls of Generalized Roth-Lempel Codes and Their Applications to EAQECCs
BibTeX
@misc{mezinaj2026hulls,
title = {{Hulls, linear equivalence, and weighted superelliptic codes}},
author = {Jurgen Mezinaj and Tanush Shaska},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.02850},
eprint = {2608.02850},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02850},
}
RIS
TY - UNPB
TI - Hulls, linear equivalence, and weighted superelliptic codes
AU - Jurgen Mezinaj
AU - Tanush Shaska
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2608.02850
UR - https://arxiv.org/abs/2608.02850
AB - The containment of the code of the meet G∧ A in the hull and the identity G∨ A−D=K−G∧ A exchanging meet and join are known; imposing that G∧ A be principal constructs algebraic geometry codes with one-dimensional hull. We turn that construction into a measurement. For arbitrary divisors G and A we compute C_L(D,G)∩ C_L(D,A) exactly: it is the code of the meet together with an excess ε(G,A), canonically their quotient and a subquotient of H^1(𝒪(G∧ A)). So ε vanishes exactly when the meet is non-special; otherwise it certifies that K−G∧ A is linearly equivalent to an effective divisor, at degree zero the vanishing of a single class in the Picard group: the hull detects a linear equivalence rather than being built from one. For superelliptic curves y^n=f(x) both sides can be computed: their weighted plane models in ℙ^2_(1,n/c,d/c), c=gcd(n,d), identify codes C_s of weighted forms of degree s with those of sD_∞ and turn hulls into lattice counts. The range on which ε is blind is an explicit interval of degrees, where dimHull(C_s)=cμ(s)−nδ+1−g_X, μ(s)=min{s,M−s}, depends only on its affine-point count. Outside it the meet and join are invariant under s↦ M−s while ε is not, so every asymmetry of the hull profile is excess and the threshold in s refines the divisor class: two totally split curves of genus two, over 𝔽_7 and over 𝔽_11, present the same class at the same pair of degrees and are separated by the profile alone. If 0≤°(G∧ A)≤2g_X−2 the hull is at most g_X+1, so it is large only where it is blind, and over a prime field, under an explicit inequality on (n,d,q), its maximum over the family is ℓ(⌊ M/2⌋ D_∞), attained exactly on the totally split locus.
ER -