Quantum Cyclic Code

$4^m+1$, $m$ any positive integer, over $\mathbb{F}_2$. We give a
complete characterisation of this family of codes and give
polynomial time quantum algorithms for decoding. More precisely, we
can extend the notion of BCH distance of classical cyclic codes to
these codes and correct errors within the BCH limit, using an
appropriate generalisation of the Berlekamp's algorithm.
Previously, such algorithms were known only for cyclic codes
constructed using the CSS constructions, where as, the codes that we
construct are not CSS.

In order to construct these codes, we start from a general notion of
cyclicity for quantum code. This can be of separate interest as our
notion of cyclicity is applicable to non-stabiliser codes as well.