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.