C. Padró and I. Gracia
Representing small identically
self-dual matroids
by self-dual codes
SIAM Journal on Discrete
Mathematics 20 (2006)
1046-1055.
Abstract
The matroid associated to a linear code is the representable
matroid that is defined by the columns of any generator matrix. The
matroid associated to 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 CDGJLMP05,
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.