Chapter 1 Quantum error-correcting codes from algebraic curves

This chapter discusses quantum error-correcting codes constructed from algebraic curves. We give an introduction to quantum coding theory including bounds on quantum codes. We describe stabilizer codes which are the quantum analog of classical linear codes and discuss the binary and q-ary CSS construction. Then we focus on quantum codes from algebraic curves including the projective line, Hermitian curves, and hyperelliptic curves. In addition, we describe the asymptotic behav-iors of quantum codes from the Garcia-Stichtenoth tower attaining the Drinfeld-Vlădut ̧ bound. One of the applications of algebraic geometry (AG) codes is their use in the construction of quantum error-correcting codes. Quantum error-correction was developed by Shor [31] and has become one of key ingre-dients in quantum computation and quantum information theory. Calder