A set Sβ{0,1}E of binary vectors, with positions indexed by E,
is said to be a \textit{powerful code} if, for all XβE, the number
of vectors in S that are zero in the positions indexed by X is a power of
2. By treating binary vectors as characteristic vectors of subsets of E, we
say that a set Sβ2E of subsets of E is a \textit{powerful set} if
the set of characteristic vectors of sets in S is a powerful code. Powerful
sets (codes) include cocircuit spaces of binary matroids (equivalently, linear
codes over F2β), but much more besides. Our motivation is that, to
each powerful set, there is an associated nonnegative-integer-valued rank
function (by a construction of Farr), although it does not in general satisfy
all the matroid rank axioms.
In this paper we investigate the combinatorial properties of powerful sets.
We prove fundamental results on special elements (loops, coloops, frames,
near-frames, and stars), their associated types of single-element extensions,
various ways of combining powerful sets to get new ones, and constructions of
nonlinear powerful sets. We show that every powerful set is determined by its
clutter of minimal nonzero members. Finally, we show that the number of
powerful sets is doubly exponential, and hence that almost all powerful sets
are nonlinear.Comment: 19 pages. This work was presented at the 40th Australasian Conference
on Combinatorial Mathematics and Combinatorial Computing (40ACCMCC),
University of Newcastle, Australia, Dec. 201