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arXiv:2101.06461 · IEEE Transactions on Information Theory

On Linear Codes With One-Dimensional Euclidean Hull and Their Applications to EAQECCs

Lin Sok

Published Feb 23, 2022• 44 citations
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Abstract

The Euclidean hull of a linear code$C$is the intersection of$C$with 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.

DOI
10.1109/tit.2022.3152580

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IEEE Transactions on Information Theory•Oct 6, 2021•49 citations
The Euclidean hull of a linear codeCCCis the intersection ofCCCwith its Euclidean dualC⊥C^\perp C⊥. 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.
The intersection C⋂C⊥{\bf C}\bigcap {\bf C}^{\perp}C⋂C⊥ (C⋂C⊥h{\bf C}\bigcap {\bf C}^{\perp_h}C⋂C⊥h​) of a linear code C{\bf C}C and its Euclidean dual C⊥{\bf C}^{\perp}C⊥ (Hermitian dual C⊥h{\bf C}^{\perp_h}C⊥h​) is called the Euclidean (Hermitian) hull of this code. It is natural to consider the hull-variation problem when a linear code C{\bf C}C is transformed to an equivalent code v⋅C{\bf v} \cdot {\bf C}v⋅C. In this paper we introduce the maximal hull dimension as an invariant of a linear code with respect to the equivalent transformations. Then some basic properties of the maximal hull dimension are studied. We prove that for a nonnegative integer hhh satisfying 0≤h≤n−10 \leq h \leq n-10≤h≤n−1, a linear [2n,n]q[2n, n]_q[2n,n]q​ self-dual code is equivalent to a linear hhh-dimension hull code. On the opposite direction we prove that a linear LCD code over F2s{\bf F}_{2^s}F2s​ satisfying d≥2d\geq 2d≥2 and d⊥≥2d^{\perp} \geq 2d⊥≥2 is equivalent to a linear one-dimension hull code under a weak condition. Several new families of LCD negacyclic codes and LCD BCH codes over F3{\bf F}_3F3​ are also constructed. Our method can be applied to the generalized Reed-Solomon codes and the generalized twisted Reed-Solomon codes to construct arbitrary dimension hull MDS codes. Some new entanglement-assisted quantum error-correction (EAQEC) codes including MDS and almost MDS EAQEC codes are constructed. Many EAQEC codes over small fields are constructed from optimal Hermitian self-dual codes.
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​​⌉.
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 $-dimension Hermitian hull of a smaller length. This construction includes the converse of the famous shortening technique as a special case. Using this method, we construct optimal quaternary linear codes of lengths up to 13 with Hermitian hull dimensions 2-5. As an application, we construct many EAQECCs, which improve the parameters of EAQECCs of Grassl's code table.
In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, where we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several new families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. Moreover, several new classes of q-ary MDS EAQECCs of length n > q+1 are also obtained.
Letq=peq=p^{e}q=pebe a prime power and$\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.