New q-ary quantum MDS codes of length strictly larger than q+1

Quantum information and quantum computation have become a hot topic in recent decades. Quantum error-correcting codes are useful and have many applications in quantum computations and quantum communications. We construct a new class of quantum Maximum Distance Separable (MDS) codes. Our construction is based on a recent result of Ball and Vilar (IEEE Trans Inf Theory 68:3796-3805, 2022). We study a large class of explicit polynomials and obtain their required arithmetical properties which imply construction of new q-ary quantum MDS codes of length strictly larger than q+1, when q is odd.