The Bush matrix over a Galois field and error correcting quantum codes

Abstract

Using the method of Bush in the construction of orthogonal arrays [A.S. Hedayat, N.J.A. Sloane, J. Stufken, Orthogonal Arrays; Theory and Applications, Springer Series in Statistics, Springer, Berlin, 1999] and the theory of characters of finite abelian groups we construct a family of error correcting quantum codes. The trade-off between the dimension of the quantum code and the number of errors corrected is investigated in this class. Associated with each prime we present an explicit family of error correcting quantum codes. Our proofs depend on the well-known Knill-Laflamme criterion [E. Knill, R. Laflamme, Phys. Rev. A 55 (1997) 900] for error correction and a basic result of A.R. Calderbank et al. [IEEE Trans. Inform. Theory 44 (1998) 1369] modified appropriately to the context of finite abelian groups.