MDS Codes, Secret Sharing, and Invariant Theory
Aiden A. Bruen, Mario Professor Forcinito, James Professor McQuillan · 2021
In this chapter, the authors provide a quick summary of some nonlinear Maximum Distance Separable (MDS) codes and their applications to secret sharing schemes and combinatorics. The MacWilliams identities are presented and used in connection with ideas from invariant theory applied to linear codes related to projective planes. In particular, the authors discuss the “computer algebra theorem of the twentieth century” – namely the nonexistence of a projective plane of order 10. Bruen showed that the existence of large unembeddable nets close to the Bruck bound using a centuries-old geometrical result known as Galluci’s Theorem or the Theorem of Dandelin.