Burst error-correcting quantum stabilizer codes designed from idempotents

Abstract

AbstractCertain 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 $$F_2$$ F 2 -submodules over GF(4). Therefore, this paper provides a classification on idempotents in commutative group algebra GF(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 $$C_n$$ C n , for $$n=2^m-1$$ n = 2 m - 1 and $$n=2^m+1$$ n = 2 m + 1 . Quantum bounds on their burst error minimum distance are subsequently determined.