  Stabilizer codes form an important class of quantum error correcting codes
which have an elegant theory, efficient error detection, and many known
examples. Constructing stabilizer codes of length $n$ is equivalent to
constructing subspaces of $\mathbb{F}_p^n \times \mathbb{F}_p^n$ which are
"isotropic" under the symplectic bilinear form defined by $\left\langle
(\mathbf{a},\mathbf{b}),(\mathbf{c},\mathbf{d}) \right\rangle =
\mathbf{a}^{\mathrm{T}} \mathbf{d} - \mathbf{b}^{\mathrm{T}} \mathbf{c}$. As a
result, many, but not all, ideas from the theory of classical error correction
can be translated to quantum error correction. One of the main theoretical
contribution of this article is to study stabilizer codes starting with a
different symplectic form.
  In this paper, we concentrate on cyclic codes. Modifying the symplectic form
allows us to generalize the previous known construction for linear cyclic
stabilizer codes, and in the process, circumvent some of the Galois theoretic
no-go results proved there. More importantly, this tweak in the symplectic form
allows us to make use of well known error correcting algorithms for cyclic
codes to give efficient quantum error correcting algorithms. Cyclicity of error
correcting codes is a "basis dependent" property. Our codes are no more
"cyclic" when they are derived using the standard symplectic forms (if we
ignore the error correcting properties like distance, all such symplectic forms
can be converted to each other via a basis transformation). Hence this change
of perspective is crucial from the point of view of designing efficient
decoding algorithm for these family of codes. In this context, recall that for
general codes, efficient decoding algorithms do not exist if some widely
believed complexity theoretic assumptions are true.
