Quantum Codes and Abelian Subgroups of the Extra-Special Group

A method for constructing a new class of quantum codes is proposed. In the method, properties of the extraspecial matrix group are exploited. This class of codes is wider than that considered in [1–4] (CSS-codes). In particular, this class includes the one-error-correcting quantum “Hamming code” (which is not a CSS-code) of length $n=2^m$, the number of elements in it being close to the maximum possible. The latter result is one of the main results of the paper.