Quantum codes and Abelian subgroups of the extra-special group (Q1812357)

Let \(F_2\) be the two element field, let \(n\) be a positive integer and let \(L\) be a subspace of the vector space \(F^n_2\). Denote by \(L^c\) any of its complements (i.e., \(F^n_2= L\oplus L^c\)). Given a subspace \(L\) and a subset \(\Gamma\subseteq (L^\perp)^c\times L^c\), the author constructs a quantum code \(Q_L(\Gamma)\). He obtains necessary and sufficient conditions for \(Q_L(\Gamma)\) to have code distance at least \(d\). The class of quantum codes \(Q_L(\Gamma)\) is wider than the class of CSS-codes studied in [\textit{A. M. Steane}, Phys. Rev. Lett. 77, 793--797 (1996; Zbl 0944.81505) and \textit{A. R. Calderbank}, \textit{E. M. Rains}, \textit{P. W. Shor} and \textit{N. J. A. Sloane}, IEEE Trans. Inf. Theory 44, 1369--1387 (1998; Zbl 0982.94029)]. The author also constructs a family of one-error-correcting quantum Hamming codes of length \(n= 2^m\) and dimension \(2^{n-m-2}\).