Quantum Codes from Two-Point Divisors on Hermitian and Suzuki Curves

Sarvepalli and Klappenecker showed how to use classical one-point codes on the Hermitian curve to construct symmetric quantum codes. For the said curve, Homma and Kim determined the parameters of a larger family of codes, the two-point codes. For the Suzuki curve, the bound due to Duursma and Kirov gave the exact minimum distance of Suzuki two-point codes over F8 and F32. Of more recent development in quantum error-correction, the observed presence of asymmetry in some binary quantum channels led to the mathematical study of asymmetric quantum codes (AQCs) where the model no longer assumes that the different types of errors are equiprobable. This paper considers quantum codes, both symmetric and asymmetric, constructed from two-point divisors on Hermitian and Suzuki curves based on the CSS construction. In the asymmetric case, we show strict improvements over all suitable finite fields for a range of designed distances. We produce large dimensions pure AQCs and small dimensions impure AQCs that have better parameters than the best-possible AQCs from one-point codes. Exact numerical results for Hermitian curves HF q2 with q ∈ {3, 4, 5, 7, 8} and for Suzuki curves SFq with q ∈ {8, 32} are used to illustrate the gain.