Burst error-correcting quantum stabilizer codes designed from idempotents

Certain classical codes can be viewed isomorphically as ideals of group algebras, while studying their algebraic structures help extracting the code properties. Research has shown that this was remarkably efficient in the case when the code generators are idempotents. In quantum error correction, the theory of stabilizer formalism requires classical self-orthogonal additive codes over the finite field GF(4), which, via the lens of group algebras, are essentially F2-submodules over GF(4). Therefore, this paper provides a classification on idempotents in commutative group algebra G F(4)G, followed by a criterion that allows idempotents to generate stabilizer subgroups. Later, the construction of quantum stabilizer codes is done in the case when G is a cyclic group Cn, for n = 2(m) - 1 and n = 2(m) + 1. Quantum bounds on their burst error minimum distance are subsequently determined.

Automated summary

The algebraic structures of classical codes can extract their properties and be viewed as ideals of group algebras. It has been proven that this method is efficient when the code generators are idempotents. In quantum error correction, self-orthogonal additive codes over GF(4) are required for the stabilizer formalism, which can be seen as F2-submodules over GF(4) through group algebras. This paper classifies idempotents in the commutative group algebra GF(4)G, and provides a criterion for idempotents to generate stabilizer subgroups. Furthermore, it constructs quantum stabilizer codes for cyclic group Cn (n = 2(m) - 1 and n = 2(m) + 1) and determines the quantum bounds on their burst error minimum distance.