  We propose two types, namely Type-I and Type-II, quantum stabilizer codes
using quadratic residue sets of prime modulus given by the form $p=4n\pm1$. The
proposed Type-I stabilizer codes are of cyclic structure and code length $N=p$.
They are constructed based on multi-weight circulant matrix generated from
idempotent polynomial, which is obtained from a quadratic residue set. The
proposed Type-II stabilizer codes are of quasi-cyclic (QC) structure and code
length $N=pk$, where $k$ is the size of a quadratic residue set. They are
constructed based on structured sparse-graphs codes derived from proto-matrix
and circulant permutation matrix. With the proposed methods, we design rich
classes of cyclic and quasi-cyclic quantum stabilizer codes with variable code
length. We show how the commutative constraint (also referred to as the
Symplectic Inner Product constraint) for quantum codes can be satisfied for
each proposed construction method. We also analyze both the dimension and
distance for Type-I stabilizer codes and the dimension of Type-II stabilizer
codes. For the cyclic quantum stabilizer codes, we show that they meet the
existing distance bounds in literature.
