Quantum stabilizer codes from Abelian and non-Abelian groups association schemes

Abstract

A new method for the construction of the binary quantum stabilizer codes is provided, where the construction is based on Abelian and non-Abelian groups association schemes. The association schemes based on non-Abelian groups are constructed by bases for the regular representation from U6n, T4n, V8n and dihedral D2n groups. By using Abelian group association schemes followed by cyclic groups and non-Abelian group association schemes a list of binary stabilizer codes up to 40 qubits is given in tables 4, 5 and 10. Moreover, several binary stabilizer codes of minimum distances 5, 7 and 8 with good quantum parameters is presented. The preference of this method specially for Abelian group association schemes is that one can construct any binary quantum stabilizer code with any distance by using the commutative structure of association schemes.