Lagrange interpolation polynomials and generalized Reed-Solomon codes over rings of matrices

In this paper we first give the Lagrange interpolation polynomials in rings of matrices over finite fields and propose a new secret sharing scheme similar to Shamir´s secret sharing scheme as its direct application in cryptography. We also introduce the generalized Reed-Solomon codes over a noncommutative ring of matrices over a finite field and study their properties using this interpolation formula. We show that they are maximum distance separable (MDS) and present the key equation by deriving their dual codes and Goppa formulation.