Minimal blocks of binary even-weight vectors

Odd circuits are minimal 1-blocks over GF(2) and the odd circuit of size 2t + 1 can be represented by the vectors of Hamming weight 2t in a (2t + 1)-dimensional vector space over GF(2). This is the tip of an iceberg. Let f(2t, k, 2) be the maximum number of binary k-dimensional column vectors such that for all s, 1 ⩽ s ⩽ t, no 2s columns sum to the zero vector. If k = 2, k = 3, k = 4, or k ⩾ 5 and 2t is sufficiently large (for example, 2t ⩾ 2k − k + 1 suffices), then the set of vectors of weight 2t in a (f(2t, k, 2) + 2t −1)-dimensional vector space over GF(2) is a minimal k-block over GF(2).