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IEEE Information Theory Workshop (ITW)

On Two-Point Rational AG Codes with Lower Bounded Hermitian Hull Dimension

Lin Sok, San Ling

Published Sep 29, 2025• 0 citations
Publisher

Abstract

Motivated by the Hermitian construction of quantum codes using classical linear codes as ingredients, we study the Hermitian hulls of two-point rational algebraic geometry (AG) codes Cℒ(D,G) with G = (k−1)O+rP for some positive integers k and r. We provide a lower bound on their Hermitian hull dimensions. Specifically, we prove that under certain conditions, the Hermitian hull dimensions of such AG codes are lower bounded by k−1−r, giving rise to linear codes of Hermitian hull dimension k−r−1 by some propagation rules. As an application, we derive two families of entanglement-assisted quantum error-correcting codes with new parameters.

DOI
10.1109/itw62417.2025.11240398

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Motivated by the Hermitian construction of quantum codes using classical linear codes as ingredients, we study the Hermitian hulls of two-point rational algebraic geometry (AG) codes Cℒ(D,G) with G = (k−1)O+rP for some positive integers k and r. We provide a lower bound on their Hermitian hull dimensions. Specifically, we prove that under certain conditions, the Hermitian hull dimensions of such AG codes are lower bounded by k−1−r, giving rise to linear codes of Hermitian hull dimension k−r−1 by some propagation rules. As an application, we derive two families of entanglement-assisted quantum error-correcting codes with new parameters.
We prove that any Hermitian self-orthogonal [n,k,d]q2[n,k,d]_{q^2}[n,k,d]q2​ code gives rise to an [n,k,d]q2[n,k,d]_{q^2}[n,k,d]q2​ code with ℓ\ellℓ dimensional Hermitian hull for 0≤ℓ≤k0\le \ell \le k0≤ℓ≤k. We present a new method to construct Hermitian self-orthogonal [n,k]q2[n,k]_{q^2}[n,k]q2​ codes with large dimensions k>n+q−1q+1k>\frac{n+q-1}{q+1}k>q+1n+q−1​. New families of Hermitian self-orthogonal codes with good parameters are obtained; more precisely those containing almost MDS codes. By applying a puncturing technique to Hermitian self-orthogonal codes, MDS [n,k]q2[n,k]_{q^2}[n,k]q2​ linear codes with Hermitian hull having large dimensions k>n+q−1q+1k>\frac{n+q-1}{q+1}k>q+1n+q−1​ are also derived. New families of MDS, almost MDS and optimal codes with arbitrary Hermitian hull dimensions are explicitly constructed from algebraic curves. As an application, we provide entanglement-assisted quantum error correcting codes with new parameters.
Hermitian hulls of linear codes are interesting for theoretical and practical reasons alike. In terms of recent application, linear codes whose hulls meet certain conditions have been utilized as ingredients to construct entanglement-assisted quantum error correcting codes. This family of quantum codes is often seen as a generalization of quantum stabilizer codes. Theoretically, compared with the Euclidean setup, the Hermitian case is much harder to deal with. Hermitian hulls of MDS linear codes with low dimensions have been explored, mostly from generalized Reed-Solomon codes. Characterizing Hermitian hulls which themselves are MDS appears to be more involved and has not been extensively studied. This paper introduces some tools to study linear codes whose Hermitian hulls are MDS. Using the tools, we then propose explicit constructions of such codes. We consider Hermitian hulls of both Reed-Solomon and non Reed-Solomon types of linear MDS codes. We demonstrate that, given the same Hermitian hull dimensions, the codes from our constructions have dimensions which are larger than those in the literature.
Hermitian hulls of linear codes are interesting for theoretical and practical reasons alike. In terms of recent application, linear codes whose hulls meet certain conditions have been utilized as ingredients to construct entanglement-assisted quantum error correcting codes. This family of quantum codes is often seen as a generalization of quantum stabilizer codes. Theoretically, compared with the Euclidean setup, the Hermitian case is much harder to deal with. Hermitian hulls of MDS linear codes with low dimensions have been explored, mostly from generalized Reed-Solomon codes. Characterizing Hermitian hulls which themselves are MDS appears to be more involved and has not been extensively studied. This paper introduces some tools to study linear codes whose Hermitian hulls are MDS. Using the tools, we then propose explicit constructions of such codes. We consider Hermitian hulls of both Reed-Solomon and non Reed-Solomon types of linear MDS codes. We demonstrate that, given the same Hermitian hull dimensions, the codes from our constructions have dimensions which are larger than those in the literature.
The hull of a linear code is the intersection of the code and its dual code, which is effective for determining parameters of entanglement-assisted quantum error-correcting codes (EAQECCs). There are few constructions of linear codes with various Hermitian hull dimensions, aside from Hermitian LCD and self-orthogonal codes. The object of this paper is to introduce a building-up construction for constructing linear $ [n+2, k+1] $ codes with $ \ell $ or $ (\ell+1) $-dimensional Hermitian hull from a given linear $ [n, k] $ code with $ \ell $-dimension Hermitian hull of a smaller length. This construction includes the converse of the famous shortening technique as a special case. Using this method, we construct optimal quaternary linear codes of lengths up to 13 with Hermitian hull dimensions 2-5. As an application, we construct many EAQECCs, which improve the parameters of EAQECCs of Grassl's code table.
Letq=peq=p^{e}q=pebe a prime power and$\ell $be an integer with0≤ℓ≤e−10\leq \ell \leq e-10≤ℓ≤e−1. The$\ell −GaloishullofclassicallinearcodesisageneralizationoftheEuclideanhullandHermitianhull.WeprovideanecessaryandsufficientconditionunderwhichacodewordofaGRScodeoranextendedGRScodebelongstoits-Galois hull of classical linear codes is a generalization of the Euclidean hull and Hermitian hull. We provide a necessary and sufficient condition under which a codeword of a GRS code or an extended GRS code belongs to its−GaloishullofclassicallinearcodesisageneralizationoftheEuclideanhullandHermitianhull.WeprovideanecessaryandsufficientconditionunderwhichacodewordofaGRScodeoranextendedGRScodebelongstoits\ell −Galoisdualcode,generalizingboththeEuclideancaseandHermitiancaseintheliterature.Byusingfourdifferenttools:1)thenormmappingfrom-Galois dual code, generalizing both the Euclidean case and Hermitian case in the literature. By using four different tools: 1) the norm mapping from−Galoisdualcode,generalizingboththeEuclideancaseandHermitiancaseintheliterature.Byusingfourdifferenttools:1)thenormmappingfrom\mathbb {F}_{q}^{\ast }tototo\mathbb {F}_{p^{\ell }}^{\ast };2)thedirectproductoftwocyclicsubgroups;3)thecosetdecompositionofacyclicgroup;4)anadditivesubgroupof; 2) the direct product of two cyclic subgroups; 3) the coset decomposition of a cyclic group; 4) an additive subgroup of;2)thedirectproductoftwocyclicsubgroups;3)thecosetdecompositionofacyclicgroup;4)anadditivesubgroupof\mathbb {F}_{q}anditscosets,weconstructelevenfamiliesofand its cosets, we construct eleven families ofanditscosets,weconstructelevenfamiliesofq−aryMDScodeswith-ary MDS codes with−aryMDScodeswith\ell −Galoishullsofarbitrarydimensions,andgivetherelatedelevenfamiliesof-Galois hulls of arbitrary dimensions, and give the related eleven families of−Galoishullsofarbitrarydimensions,andgivetherelatedelevenfamiliesof[[n,k,d;c]]_{q}$entanglement-assisted quantum error-correcting codes (EAQECCs) with relatively large minimum distance in the sense that2d=n−k+2+c2d=n-k+2+c2d=n−k+2+c. We show that developing the theory on$\ell −Galoishullsof-Galois hulls of−Galoishullsofq−aryMDScodesinthispaperenablesustoobtainnew-ary MDS codes in this paper enables us to obtain new−aryMDScodesinthispaperenablesustoobtainnewq−aryEAQECCswithdifferentkindsoflengthsetsviadifferent-ary EAQECCs with different kinds of length sets via different−aryEAQECCswithdifferentkindsoflengthsetsviadifferent\ell $, where2ℓ∣e2\ell \mid e2ℓ∣e.