On the rank of a random binary matrix

We study the rank of a random nxm matrix A(n,m;k) with entries from GF(2), and exactly k unit entries in each column, the other entries being zero. The columns are chosen independently and uniformly at random from the set of all ((n)(k)) such columns. We obtain an asymptotically correct estimate for the rank as a function of the number of columns m in terms of c, n, k, and where m = cn/k. The matrix A(n,m;k) forms the vertex-edge incidence matrix of a k-uniform random hypergraph H. The rank of A(n,m;k) can be expressed as follows. Let vertical bar C-2 vertical bar be the number of vertices of the 2-core of H, and vertical bar E(C-2)vertical bar the number of edges. Let m* be the value of m for which vertical bar C-2 vertical bar = vertical bar E(C-2)vertical bar. Then w.h.p. for m < m* the rank of A(n,m;k) is asymptotic to m, and for m >= m* the rank is asymptotic to m - vertical bar E(C-2)vertical bar + vertical bar(C)2 vertical bar. In addition, assign i.i.d. U[0, 1] weights X-i, i is an element of {1, 2, ..., m} to the columns, and define the weight of a set of columns S as X(S) = Sigma(j)(is an element of S) X-j. Define a basis as a set of n - 1(k even) linearly independent columns. We obtain an asymptotically correct estimate for the minimum weight basis. This generalises the well-known result of Frieze [On the value of a random minimum spanning tree problem, Discrete Applied Mathematics, (1985)] that, for k = 2, the expected length of a minimum weight spanning tree tends to zeta(3) similar to 1.202.