Construct longer quantum MDS codes with large minimum distance

Construct quantum maximum-distance-separable codes of lengths greater than q+1 whose minimum distances exceed q/2+1, where q is an odd prime power.

Background

Quantum MDS codes attain the quantum Singleton bound, and the Hermitian construction reduces their construction to finding Hermitian self-orthogonal classical MDS codes over finite fields. Existing constructions cover broad parameter ranges, but the attainable lengths and distances become more restrictive when the length exceeds q+1.

The paper develops five families of quantum MDS codes using complements of unions of structured subsets of F_{q^2}. These constructions provide codes with lengths of the form q^2-Sq+theta and, under suitable conditions, minimum distances greater than q/2+1. Thus, the paper addresses the broader unresolved construction problem by supplying new families rather than establishing a complete characterization of all such lengths and distances.