We consider linear secret sharing schemes (LSSS) over a finite field K with the shares in K. An LSSS with t-adversary and n players is strongly multiplicative if it has (n−t)-product reconstruction...Show moreWe consider linear secret sharing schemes (LSSS) over a finite field K with the shares in K. An LSSS with t-adversary and n players is strongly multiplicative if it has (n−t)-product reconstruction. It is well-known that for strongly multiplicative LSSS with the secret in K it holds that t ≤ n−1 3 . This bound is sharp, as equality can be attained using Shamir’s scheme. We show that in fact Shamir’s scheme is the only strongly multiplicative LSSS with maximal adversary t. We generalize this result to strongly multiplicative LSSS with the secret in an extension field L over K of finite degree k. We show that it holds that t ≤ n−2k+1 3 , and that equality can be attained using an extension of Shamir’s scheme, where we take the evaluation point of the secret in L. We also show that this scheme is the only one that attains maximal t. We build on earlier work by Mirandola and Z´emor from 2015, who showed a coding-theoretic version of Vosper’s theorem, a classical result from additive combinatorics. This theorem states in particular that a linear MDS code C of length n is Reed-Solomon if the dimension of its Schur square C ∗2 satisfies 2 < dim C ∗2 = 2 dim C − 1 < n − 1. We discuss whether this theorem also applies to non-MDS linear codes, and in doing so we provide a slight generalization of the theorem. We also prove that non-MDS codes C exist with dim C ∗2 = 2 dim C − 1 and with C of arbitrary codimension, using the amalgamated direct sum of codes. As a second coding-theoretic application of the analogue of Vosper’s theorem, we show an implication for error-correcting pairs. It was shown by M´arquez-Corbella and Pellikaan in 2016 that existence of a t-error correcting pair for an MDS code C implies that C is Reed-Solomon. They gave two separate proofs. Besides their original proof, they gave a second proof that indirectly uses the analogue of Vosper’s theorem. We show an alternative proof directly from this theorem.Show less
