On the representability of the biuniform matroid - Ball, Simeon (2014)
Every biuniform matroid is representable over all suﬃciently large ﬁelds. But it is not known exactly over which ﬁnite ﬁelds they are representable, and the existence of eﬃcient methods to ﬁnd a representation for every given biuniform matroid has not been proved. The interest of these problems is due to their implications to secret sharing. The existence of eﬃcient methods to ﬁnd representations for all biuniform matroids is proved here for the ﬁrst time. The previously known eﬃcient constructions apply only to a particular class of biuniform matroids, while the known general constructions were not proved to be eﬃcient. In addition, our constructions provide in many cases representations over smaller ﬁnite ﬁelds.