Error-correcting codes and matroids have been widely used in the study of ordinary secret sharing schemes. In this paper, the connections between codes, matroids, and a special class of secret sharing schemes, namely, multiplicative linear secret sharing schemes (LSSSs), are studied. Such schemes are known to enable multiparty computation protocols secure against general (nonthreshold) adversaries. Two open problems related to the complexity of multiplicative LSSSs are considered in this paper. The first one deals with strongly multiplicative LSSSs. As opposed to the case of multiplicative LSSSs, it is not known whether there is an efficient method to transform an LSSS into a strongly multiplicative LSSS for the same access structure with a polynomial increase of the complexity. A property of strongly multiplicative LSSSs that could be useful in solving this problem is proved. Namely, using a suitable generalization of the well-known Berlekamp-Welch decoder, it is shown that all strongly multiplicative LSSSs enable efficient reconstruction of a shared secret in the presence of malicious faults. The second one is to characterize the access structures of ideal multiplicative LSSSs. Specifically, the considered open problem is to determine whether all self-dual vector space access structures are in this situation. By the aforementioned connection, this in fact constitutes an open problem about matroid theory, since it can be restated in terms of representability of identically self-dual matroids by self-dual codes. A new concept is introduced, the flat-partition, that provides a useful classification of identically self-dual matroids. Uniform identically self-dual matroids, which are known to be representable by self-dual codes, form one of the classes. It is proved that this property also holds for the family of matroids that, in a natural way, is the next class in the above classification: the identically self-dual bipartite matroids.

The matroid associated with a linear code is the representable matroid that is defined by the columns of any generator matrix. The matroid associated with a self‐dual code is identically self‐dual, but it is not known whether every identically self‐dual representable matroid can be represented by a self‐dual code. This open problem was proposed in [R. Cramer et al., Advances in Cryptology, Lecture Notes in Comput. Sci. 3621, Springer, New York, 2005, pp. 327–343], where it was proved to be equivalent to an open problem on the complexity of multiplicative linear secret sharing schemes. Some contributions to its solution are given in this paper. A new family of identically self‐dual matroids that can be represented by self‐dual codes is presented. Additionally, we prove that every identically self‐dual matroid on at most eight points is representable by a self‐dual code.