Xing–Ling codes, duals of their subcodes, and good asymmetric quantum codes

Abstract

A class of powerful q-ary linear polynomial codes originally proposed by Xing and Ling is deployed to construct good asymmetric quantum codes via the standard CSS construction. Our quantum codes are q-ary block codes that encode k qudits of quantum information into n qudits and correct up to [(dx - 1)/2] bit-flip errors and up to [(dz - 1)/2] phase-flip errors. In many cases where the length (q2 - q)/2 ≤ n ≤ (q2 + q)/2 and the field size q are fixed and for chosen values of dx ∈ {2, 3, 4, 5} and dz ≥ δ, where δ is the designed distance of the Xing–Ling (XL) codes, the derived pure q-ary asymmetric quantum CSS codes possess the best possible size given the current state of the art knowledge on the best classical linear block codes.