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arXiv:2212.14521 · IEEE Transactions on Information Theory

Relative Hulls and Quantum Codes

Sarah E. Anderson, Eduardo Camps-Moreno, Hiram H. López, Gretchen L. Matthews, Diego Ruano, Ivan Soprunov

Published Mar 8, 2024• 32 citations
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Abstract

Given two$q$-ary codes$C_{1}$and$C_{2}$, the relative hull of$C_{1}$with respect to$C_{2}$is the intersection$C_{1}\cap C_{2}^{\perp} $. We prove that when$q>2$, the relative hull dimension can be repeatedly reduced by one, down to a certain bound, by replacing either of the two codes with an equivalent one. The reduction of the relative hull dimension applies to hulls taken with respect to the$e$-Galois inner product, which has as special cases both the Euclidean and Hermitian inner products. We give conditions under which the relative hull dimension can be increased by one via equivalent codes when$q>2$. We study some consequences of the relative hull properties on entanglement-assisted quantum error-correcting codes and prove the existence of new entanglement-assisted quantum error-correcting maximum distance separable codes, meaning those whose parameters satisfy the quantum Singleton bound.

DOI
10.1109/tit.2024.3373550

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Given twoqqq-ary codesC1C_{1}C1​andC2C_{2}C2​, the relative hull ofC1C_{1}C1​with respect toC2C_{2}C2​is the intersectionC1∩C2⊥C_{1}\cap C_{2}^{\perp} C1​∩C2⊥​. We prove that whenq>2q>2q>2, the relative hull dimension can be repeatedly reduced by one, down to a certain bound, by replacing either of the two codes with an equivalent one. The reduction of the relative hull dimension applies to hulls taken with respect to theeee-Galois inner product, which has as special cases both the Euclidean and Hermitian inner products. We give conditions under which the relative hull dimension can be increased by one via equivalent codes whenq>2q>2q>2. We study some consequences of the relative hull properties on entanglement-assisted quantum error-correcting codes and prove the existence of new entanglement-assisted quantum error-correcting maximum distance separable codes, meaning those whose parameters satisfy the quantum Singleton bound.
The Galois hull of a linear code is the intersection of itself and its Galois dual code, which has aroused the interest of researchers in these years. In this paper, we study Galois hulls of linear codes. Firstly, the symmetry of the dimensions of Galois hulls of linear codes is found. Some new necessary and sufficient conditions for linear codes being Galois self-orthogonal codes, Galois self-dual codes, and Galois linear complementary dual codes are characterized. Then, we propose explicit methods to construct Galois self-orthogonal codes of larger length from given Galois self-orthogonal codes. As an application, linear codes of larger length with Galois hulls of arbitrary dimensions are further derived. Focusing on the Hermitian inner product, two new classes of Hermitian self-orthogonal maximum distance separable (MDS) codes are also constructed. Finally, applying all the results to the construction of entanglement-assisted quantum error-correcting codes (EAQECCs), many new qqq-ary or q\sqrt{q}q​-ary EAQECCs and MDS EAQECCs with rates greater than or equal to 12\frac{1}{2}21​ and positive net rates can be obtained. Moreover, the minimum distance of many q\sqrt{q}q​-ary MDS EAQECCs of length n>q+1n>\sqrt{q}+1n>q​+1 is greater than or equal to ⌈q2⌉\lceil \frac{\sqrt{q}}{2} \rceil⌈2q​​⌉.
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The σσσ hulls of matrix-product codes and related entanglement-assisted quantum error-correcting codes
Let SLAut(Fqn)\mathrm{SLAut}(\mathbb{F}_{q}^{n})SLAut(Fqn​) denote the group of all semilinear isometries on Fqn\mathbb{F}_{q}^{n}Fqn​, where q=peq=p^{e}q=pe is a prime power. Matrix-product (MP) codes are a class of long classical codes generated by combining several commensurate classical codes with a defining matrix. We give an explicit formula for calculating the dimension of the σσσ hull of a MP code. As a result, we give necessary and sufficient conditions for the MP codes to be σσσ dual-containing and σσσ self-orthogonal. We prove that dimFq(Hullσ(C))=dimFq(Hullσ(C⊥σ))\mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}))=\mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}^{\bot_σ}))dimFq​​(Hullσ​(C))=dimFq​​(Hullσ​(C⊥σ​)). We prove that for any integer hhh with max{0,k1−k2}≤h≤dimFq(C1∩C2⊥σ)\mathrm{max}\{0,k_{1}-k_{2}\}\leq h\leq \mathrm{dim}_{\mathbb{F}_{q}}(\mathcal{C}_{1}\cap\mathcal{C}_{2}^{\bot_σ})max{0,k1​−k2​}≤h≤dimFq​​(C1​∩C2⊥σ​​), there exists a linear code C2,h\mathcal{C}_{2,h}C2,h​ monomially equivalent to C2\mathcal{C}_{2}C2​ such that dimFq(C1∩C2,h⊥σ)=h\mathrm{dim}_{\mathbb{F}_{q}}(\mathcal{C}_{1}\cap\mathcal{C}_{2,h}^{\bot_σ})=hdimFq​​(C1​∩C2,h⊥σ​​)=h, where Ci\mathcal{C}_{i}Ci​ is an [n,ki]q[n,k_{i}]_{q}[n,ki​]q​ linear code for i=1,2i=1,2i=1,2. We show that given an [n,k,d]q[n,k,d]_{q}[n,k,d]q​ linear code C\mathcal{C}C, there exists a monomially equivalent [n,k,d]q[n,k,d]_{q}[n,k,d]q​ linear code Ch\mathcal{C}_{h}Ch​, whose σσσ dual code has minimum distance d′d'd′, such that there exist an [[n,k−h,d;n−k−h]]q[[n,k-h,d;n-k-h]]_{q}[[n,k−h,d;n−k−h]]q​ EAQECC and an [[n,n−k−h,d′;k−h]]q[[n,n-k-h,d';k-h]]_{q}[[n,n−k−h,d′;k−h]]q​ EAQECC for every integer hhh with 0≤h≤dimFq(Hullσ(C))0\leq h\leq \mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}))0≤h≤dimFq​​(Hullσ​(C)). Based on this result, we present a general construction method for deriving EAQECCs with flexible parameters from MP codes related to σσσ hulls.
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 $, where2ℓ∣e2\ell \mid e2ℓ∣e.
The Euclidean hull of a linear codeCCCis the intersection ofCCCwith its Euclidean dual$C^\perp $. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The Euclidean hull of a linear code has been applied to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs) via classical error-correcting codes. In this paper, we firstly consider linear codes with one-dimensional Euclidean hull from algebraic geometry codes, and then present a general method to construct linear codes with arbitrary dimensional Euclidean hull. Some new EAQECCs are presented.
In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, where we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several new families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. Moreover, several new classes of q-ary MDS EAQECCs of length n > q+1 are also obtained.