Quantum codes over Eisenstein-Jacobi integers

Abstract

In this study, we construct quantum error correcting codes over Eisenstein-Jacobi integers by using the CSS code construction. Since there is an isomorphism between Eisenstein- Jacobi integers and finite fields, direct constructions of quantum codes over Eisenstein-Jacobi integers can be obtained. Therefore, we define error bases, error matrices and a new distance with giving illustrative examples. Also, we prove the commutative property of error operators with respect to this new distance. Obtaining these codes can lead an answer for the existence question for some new parameters.