In this paper, we first give the definition of the Euclidean sums of linear codes, and prove that the Euclidean sums of linear codes are Euclidean dual-containing. Then we construct two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of the Reed-Solomon codes, and two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of linear codes generated by Vandermonde matrices over finite fields. Moreover, these optimal asymmetric quantum error-correcting codes constructed in this paper are different from the ones in the literature.
In recent literature, matrix product (MP) codes and their duals have gained significant attention due to their application in the construction of quantum stabilizer codes. In this paper, we begin with providing a formula that characterizes the Galois dual of MP codes. Using this formula, we establish the conditions under which MP codes are self-orthogonal and dual-containing. Although similar results may exist in the literature, the novelty and superiority of our results can be identified in the following points. Previous results that characterize the duals of MP codes only apply to MP codes with an invertible square defining matrix
A. However, our characterization applies to MP code with any defining matrix, whether
A is not square or not of full row rank. Previous studies on the conditions for self-orthogonality or dual-containment of MP codes have assumed certain structures for the product
AAT or
AA†, such as being diagonal, anti-diagonal, monomial, or partitioned Hermitian orthogonal. However, our conditions do not necessitate such specific structures. Previous studies investigated MP code duality in the context of Euclidean and Hermitian duals; however, we investigate MP code duality in the broader context of Galois dual, with Euclidean and Hermitian duals acting as special cases. Finally, it is worth noting that the proposed conditions for Galois self-orthogonality or dual-containment are both necessary and sufficient. To demonstrate the theoretical results, several numerical examples with best-known parameters MP codes are provided.
In this paper, we employ MP codes whose defining matrices are
τ-optimal defining (
τ-OD) matrices to construct new quantum codes and quantum
(r,δ)-LRCs. Specifically, we report the following results: We establish a unified
τ-monomial decomposition theorem for invertible self-adjoint matrices over finite fields of arbitrary characteristic, which generalizes the result in "Quantum codes using the
τ-OD MP construction" where the characteristic was required to be odd. Based on this theorem, we prove the existence of
τ-OD matrices over
Fq2 for any characteristic and demonstrate that there exist several new infinite families of
τ-OD matrices over
Fq2 of characteristic
2. As an application of MP codes involving
τ-OD matrices, we construct several infinite families of quantum codes with flexible parameters. Within this framework, we present
222 record-breaking quantum codes that surpass the best-known records maintained in Grassl's database.
We propose two effective schemes for constructing optimal pure quantum
(r,δ)-LRCs via MP codes. Accordingly, we construct four new infinite families of optimal pure quantum
(r,δ)-LRCs with flexible parameters. Notably, we report an interesting phenomenon by exhibiting
30 optimal pure quantum
(r,δ)-LRCs derived from our framework; that is, there exist quantum codes that are not only optimal pure quantum
(r,δ)-LRCs but also, according to Grassl's database, best-known, optimal, or record-breaking quantum codes. To the best of our knowledge, the new discovery that quantum codes are simultaneously optimal pure quantum
(r,δ)-LRCs and record-breaking quantum codes has not been previously reported in the literature.
Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good
q-ary quantum codes of large length. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in [10-12]; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and
q-ary quantum codes derived from these matrix-product codes.
We construct three new families of asymmetric quantum MDS codes from nested Hermitian self-orthogonal generalized Reed–Solomon and extended generalized Reed–Solomon codes over Fq2. The construction is developed in three settings: affine partitions of Fq2, projective norm partitions of Fq2*, and extended affine configurations obtained by adjoining the point at infinity. In each case, the Hermitian orthogonality conditions are reduced to explicit linear systems over Fq, whose solvability follows from structured moment identities and Vandermonde-type arguments. This yields nested classical MDS codes satisfying the Hermitian dual-containment condition required in the Hermitian construction of asymmetric quantum codes. As a consequence, we obtain three explicit families of asymmetric quantum MDS codes with fully determined lengths, dimensions, and asymmetric distances dz and dx. Our results show that affine and projective partition techniques provide a natural and effective framework for constructing optimal asymmetric quantum codes with flexible parameters.