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arXiv:2312.13921 · SIAM Journal on Applied Algebra and Geometry

Hulls of Projective Reed–Muller Codes over the Projective Plane

Diego Ruano, Rodrigo San–José

Published Nov 18, 2024• 6 citations
arXivHTMLPDF

Abstract

Abstract. By solving a problem regarding polynomials in a quotient ring, we obtain the relative hull and the Hermitian hull of projective Reed–Muller codes over the projective plane. The dimension of the hull determines the minimum number of maximally entangled pairs required for the corresponding entanglement-assisted quantum error-correcting code. Hence, by computing the dimension of the hull we now have all the parameters of the symmetric and asymmetric entanglement-assisted quantum error-correcting codes constructed with projective Reed–Muller codes over the projective plane. As a byproduct, we also compute the dimension of the Hermitian hull for affine Reed–Muller codes in two variables.

DOI
10.1137/23m1626153

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Abstract. By solving a problem regarding polynomials in a quotient ring, we obtain the relative hull and the Hermitian hull of projective Reed–Muller codes over the projective plane. The dimension of the hull determines the minimum number of maximally entangled pairs required for the corresponding entanglement-assisted quantum error-correcting code. Hence, by computing the dimension of the hull we now have all the parameters of the symmetric and asymmetric entanglement-assisted quantum error-correcting codes constructed with projective Reed–Muller codes over the projective plane. As a byproduct, we also compute the dimension of the Hermitian hull for affine Reed–Muller codes in two variables.
Long quantum codes using projective Reed–Muller codes are constructed. Projective Reed–Muller codes are evaluation codes obtained by evaluating homogeneous polynomials at the projective space. We obtain asymmetric and symmetric quantum codes by using the CSS construction and the Hermitian construction, respectively. We provide entanglement-assisted quantum error-correcting codes from projective Reed–Muller codes with flexible amounts of entanglement by considering equivalent codes. Moreover, we also construct quantum codes from subfield subcodes of projective Reed–Muller codes.
Abstract We study the Hermitian hull of a particular family of generalized Reed–Solomon codes. The problem of computing the dimension of the hull is translated to a counting problem in a lattice. By solving this problem, we provide explicit formulas for the dimension of the hull, which determines the minimum number required of maximally entangled pairs for the associated entanglement-assisted quantum error-correcting codes. This flexible construction allows to obtain a wide range of entanglement-assisted quantum MDS codes, as well as new parameters.
We study the Hermitian hull of a particular family of generalized Reed-Solomon codes. The problem of computing the dimension of the hull is translated to a counting problem in a lattice. By solving this problem, we provide explicit formulas for the dimension of the hull, which determines the minimum number required of maximally entangled pairs for the associated entanglement-assisted quantum error-correcting codes. This flexible construction allows to obtain a wide range of entanglement-assisted quantum MDS codes, as well as new parameters.
Determining the hull of linear codes has long been an important topic in coding theory. Recently, non-generalized Reed-Solomon (in short, non-GRS) codes have attracted extensive research interest. The (L,P)-twisted generalized Reed-Solomon (in short, (L,P)-TGRS) code, which is an extension of the generalized Reed-Solomon (GRS) code, constitutes a well-studied calss of non-GRS codes.There are numerous works focusing on the Euclidean hull of (L,P)-TGRS codes, while only a few results on the Hermitian hull of (L,P)-TGRS codes. In this paper, we focus on a class of (L,P)-TGRS codes C_k(a). By taking a special class of the vector a with length i(q-1), and analyze the parity of i and the relation between i and q+1, we divide three cases to fully determine the Hermitian hull dimension of C_k(a). As an application, we construct two classes of entanglement-assisted quantum error-correcting codes.
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.
LetRSk(a){\mathrm{ RS}}_{k}(\mathbf {a})RSk​(a)be akkk-dimensional Reed-Solomon (RS) code overFq\mathbb {F}_{q}Fq​associated witha=(α1,⋯ ,αn)\mathbf {a}=(\alpha _{1},\cdots,\alpha _{n})a=(α1​,⋯,αn​)and leth=∏i=1n(z−αi)h=\prod _{i=1}^{n}(z-\alpha _{i})h=∏i=1n​(z−αi​)be a polynomial in variablezzz. In this paper, by expressingRSk(a){\mathrm{ RS}}_{k}(\mathbf {a})RSk​(a)as anL\mathcal {L}L-construction algebraic geometry code, we completely determine the dimension of the hull${\mathrm{ RS}}_{k}(\mathbf {a})\bigcap {\mathrm{ RS}}_{k}(\mathbf {a})^{\perp} intermsofthedegreeofthederivativeofin terms of the degree of the derivative ofintermsofthedegreeofthederivativeofh$and some relevant polynomials. As applications, we explicitly determine the parameters of MDS entanglement-assisted quantum error-correcting codes constructed from RS codes, and all linear complementary dual (resp. self-dual) RS codes are also fully described.