We solve the problem of the classification ofq-ary perfect quantum codes. We prove that the only nontrivialq-ary perfect quantum codes are those with the parameters (((q2l- 1)/(q2- 1),qn- 2l,3))q. There exist no other nontrivialq-ary perfect quantum codes.

Based jointly on idempotents and modular Golomb rulers, we construct a class of nonbinary cyclic low-density parity-check (LDPC) codes. The defining parity-check matrix is a sparse circulant, on which we put two constraints: 1) the characteristic polynomial is an idempotent, 2) the nonzero elements of the first row are located on a modular Golomb ruler. We show that the second constraint forms a necessary and sufficient condition for the Tanner graph to have no cycles of length 4. The minimum distance of the code is proved equal to the column weight of the parity-check matrix plus one. A search algorithm is presented, with which we obtain some high rate codes with large minimum distances. The issue of code equivalence is also discussed. Simulation results show that the obtained codes perform well under iterative decoding.

A famous open problem in the theory of quantum error-correcting codes is whether or not the parameters of an impure quantum code can violate the quantum Hamming bound for pure quantum codes. We partially solve this problem. We demonstrate that there exists a thresholdN(d,m) such that an arbitrary((n,K,d))mquantum code must obey the quantum Hamming bound whenevern≥N(d,m). We list some values ofN(d,m) for smalldand binary quantum codes.

We explicitly construct an infinite family of asymptotically good concatenated quantum stabilizer codes where the outer codes use quantum Reed-Solomon codes and the inner codes use a set of special quantum codes. In the field of quantum error-correcting codes, this is the first time that a family of asymptotically good quantum codes is derived from bad codes. Its occurrence supplies a gap in quantum coding theory.