New asymmetric quantum generalized Reed-Solomon (RS) codes with parameters [[N = mn, K = m(2k - n + c), dz≥ d/dx≥ [d - c)]]q, where q is a prime power, 1m, k = n - d + 1, and n, d >; c + 1, c ≥ 1, m ≥ 1 are integers, are constructed in this paper. These new codes can be utilized in quantum channels having great asymmetry.
In this paper, we construct two classes of asymmetric quantum codes by using constacyclic codes. The first class is the asymmetric quantum codes with parameters [[q2 + 1, q2 + 1 - 2(t + k + 1), (2k + 2)/(2t + 2)]]q2 where q is an odd prime power, t, k are integers with [Formula: see text], which is a generalization of [J. Chen, J. Li and J. Lin, Int. J. Theor. Phys. 53 (2014) 72, Theorem 2] in the sense that we do not assume that q ≡1 ( mod 4). The second one is the asymmetric quantum codes with parameters [Formula: see text], where q ≥ 5 is an odd prime power, t, k are integers with 0 ≤ t ≤ k ≤ q - 1. The constructed asymmetric quantum codes are optimal and their parameters are not covered by the codes available in the literature.
Recently, the theory of quantum error control codes has been extended to include quantum codes over asymmetric quantum channels - qubit-flip and phase-shift errors may have equal or different probabilities. Previous work in constructing quantum error control codes has focused on code constructions for symmetric quantum channels. In this paper we establish a method to construct asymmetric quantum codes based on classical codes. We derive, once again, families of asymmetric quantum codes from classical BCH and RS codes over finite fields. Particularly, we present interesting asymmetric quantum codes based on nonprimitive narrow-sense BCH codes with parameters [[n, k, dz/dx]]qfor certain values of code lengths, dimensions, and various minimum distance. Finally, our constructions are well explained by an illustrative example.
Quantum maximum-distance-separable (MDS for short) codes are an important class of quantum codes. In this paper, by using Hermitian self-orthogonal generalized Reed-Solomon (GRS for short) codes, we construct five new classes of q-ary quantum MDS codes with minimum distance larger than q/2+1. Furthermore, the parameters of our quantum MDS code cannot be obtained from the previous constructions.
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.