Good and asymptotically good quantum codes derived from algebraic geometry

Abstract

In this paper, we construct several new families of quantum codes with good parameters. These new quantum codes are derived from (classical) t-point ($t\ge 1$) algebraic geometry (AG) codes by applying the Calderbank–Shor–Steane (CSS) construction. More precisely, we construct two classical AG codes $C_1$ and $C_2$ such that $C_1\subset C_2$, applying after the well-known CSS construction to $C_1$ and $C_2$. Many of these new codes have large minimum distances when compared with their code lengths as well as they also have small Singleton defects. As an example, we construct a family ${[[46, 2(t_2 - t_1), d]]}_{25}$ of quantum codes, where $t_1 , t_2$ are positive integers such that $1<t_1< t_2 < 23$ and $d\ge \min \{ 46 - 2t_2 , 2t_1 - 2 \}$, of length $n=46$, with minimum distance in the range $2\le d\le 20$, having Singleton defect at most four. Additionally, by applying the CSS construction to sequences of t-point (classical) AG codes constructed in this paper, we generate sequences of asymptotically good quantum codes.