Asymmetric quantum Reed-Solomon and generalized Reed-Solomon codes

Two new families of asymmetric quantum codes are constructed in this paper. The first one is derived from the Calderbank-Shor-Steane (CSS) construction applied to classical Reed-Solomon (RS) codes, providing quantum codes with parameters [[N = l(q(l) -1), K = l(q(l)-2d+c+1), d(z) >= d/d(x) >= (d-c)]](q), where q is a prime power and d > c + 1, c >= 1, l >= 1 are integers. The second family is derived from the CSS construction applied to classical generalized RS codes, generating quantum codes with parameters [[N = mn, K = m(2k-n+c), dz >= d/d(x) >= (d-c)]](q), where q is a prime power, 1 < k < n < 2k + c <= q(m), k = n - d + 1, and n, d > c + 1, c >= 1, m >= 1 are integers. Although the second proposed construction generalizes the first one, the techniques developed in both constructions are slightly different. These new codes have parameters better than or comparable to the ones available in the literature. Additionally, the proposed codes can be utilized in quantum channels having great asymmetry, that is, quantum channels in which the probability of occurrence of phase-shift errors is large when compared to the probability of occurrence of qudit-flip errors.