New Quantum Codes Derived from Group Rings

In this paper, we use the principal ideals of group rings to construct quantum codes. We use multivariate polynomial rings to represent group rings and use multivariate polynomials to represent codewords. We give a condition for multivariate polynomials such that they generate Hermitian dual-containing codes. We also find that the minimum distance of the code which is generated by a multivariate polynomial is related to the zeros distribution of the multivariate polynomial. After computer search and calculation, we get many quantum codes with good parameters over small finite fields.