Representing Small Identically Self‐Dual Matroids by Self‐Dual Codes
Carles Padró, Ignacio Gracia · SIAM Journal on Discrete Mathematics · 2006
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.