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Advances in Mathematics of Communications

On cyclic and negacyclic codes with one-dimensional hulls and their applications

Steven T. Dougherty, Serap Şahinkaya

Published Dec 6, 2022• 7 citations
Publisher

Abstract

Linear codes over finite fields with small dimensional hulls have received much attention due to their applications in cryptology and quantum computing. In this paper, we study cyclic and negacyclic codes with one-dimensional hulls. We determine precisely when cyclic and negacyclic codes over finite fields with one-dimensional hulls exist. We also introduce one-dimensional linear complementary pairs of cyclic and negacyclic codes. As an application, we obtain numerous optimal or near optimal cyclic codes with one-dimensional hulls over different fields and, by using these codes, we present new entanglement-assisted quantum error-correcting codes (EAQECCs). In particular, some of these EAQEC codes are maximal distance separable (MDS). We also obtain one-dimensional linear complementary pairs of cyclic codes, which are either optimal or near optimal.

DOI
10.3934/amc.2022096

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Linear codes over finite fields with small dimensional hulls have received much attention due to their applications in cryptology and quantum computing. In this paper, we study cyclic and negacyclic codes with one-dimensional hulls. We determine precisely when cyclic and negacyclic codes over finite fields with one-dimensional hulls exist. We also introduce one-dimensional linear complementary pairs of cyclic and negacyclic codes. As an application, we obtain numerous optimal or near optimal cyclic codes with one-dimensional hulls over different fields and, by using these codes, we present new entanglement-assisted quantum error-correcting codes (EAQECCs). In particular, some of these EAQEC codes are maximal distance separable (MDS). We also obtain one-dimensional linear complementary pairs of cyclic codes, which are either optimal or near optimal.
Linear codes with small hulls over finite fields have been extensively studied due to their practical applications in computational complexity and information protection. In this paper, we develop a general method to determine the exact value of D4H(n,k,1)D_4^H(n,k,1)D4H​(n,k,1) for n≤12n\leq 12n≤12 or k∈{1,2,3,n−1,n−2,n−3}k\in \{1,2,3,n-1,n-2,n-3\}k∈{1,2,3,n−1,n−2,n−3}, where D4H(n,k,1)D_4^H(n,k,1)D4H​(n,k,1) denotes the largest minimum distance among all quaternary linear [n,k][n,k][n,k] codes with one-dimensional Hermitian hull. As a consequence, we solve a conjecture proposed by Mankean and Jitman on the largest minimum distance of a quaternary linear code with one-dimensional Hermitian hull. As an application, we construct some binary entanglement-assisted quantum error-correcting codes (EAQECCs) from quaternary linear codes with one-dimensional Hermitian hull. Some of these EAQECCs are optimal codes, and some of them are better than previously known ones.
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.
The Hermitian hull of linear codes plays an important role in coding theory and quantum coding theory. In this paper, we first construct some infinite classes of MDS codes over finite field by considering generalized Reed-Solomon codes or extended generalized Reed- Solomon codes and determine their Hermitian hulls. The results indicate that these codes constructed have Hermitian hulls of (almost) arbitrary dimensions. Furthermore, based on these MDS codes constructed with Hermitian hulls of arbitrary dimensions, we obtain several infinite classes of entanglement-assisted quantum error-correcting (EAQEC) MDS codes whose maximally entangled states c can take almost all values. Moreover, most of the EAQEC MDS codes constructed are new in the sense that their parameters are different from all the previously known ones.
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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.