On Bounds for Quantum Error Correcting Codes over EJ-Integers

Abstract

There are some differences between quantum and classical error corrections [Nielsen M.A., and I.L. Chuang, “Quantum Computation and Quantum Information”, Cambridge University Press, Cambridge, 2002.]. Hence, these differences should be considered when a new procedure is performed. In our recent study, we construct new quantum error correcting codes over different mathematical structures. The classical codes over Eisenstein-Jacobi(EJ) integers are mentioned in [Huber, K., “Codes over Eisenstein-Jacobi integers”, Contemporary Mathematics 168 (1994), 165.]. There is an efficient algorithm for the encoding and decoding procedures of these codes [Huber, K., “Codes over Eisenstein-Jacobi integers”, Contemporary Mathematics 168 (1994), 165.]. For coding over two-dimensional signal spaces like QAM signals, block codes over these integers p = 7, 13, 19, 31, 37, 43, 61, … can be useful [Dong, X., C.B. Soh, E. Gunawan and L. Tang, “Groups of Algebraic Integers used for Coding QAM Signals”, Information Theory, IEEE 44 (1998), 1848–1860.]. Thus, in this study, we introduce quantum error correcting codes over EJ-integers. This type of quantum codes may lead to codes with some new and good parameters.