We classify all q-ary Δ-divisible linear codes which are spanned by codewords of weight Δ. The basic building blocks are the simplex codes, and for q=2 additionally the first order Reed-Muller codes and the parity check codes. This generalizes a result of Pless and Sloane, where the binary self-orthogonal codes spanned by codewords of weight 4 have been classified, which is the case q=2 and Δ=4 of our classification. As an application, we give an...

A vector space partition P of the projective space PG(v-1,q) is a set of subspaces in PG(v-1,q) which partitions the set of points. We say that a vector space partition P has type (v-1)^{m_{v-1}} ... 2^{m_2}1^{m_1} if precisely m_i of its elements have dimension i, where 1 <= i <= v-1. Here we determine all possible types of vector space partitions in PG(7,2).

A subcube partition is a partition of the Boolean cube {0,1}^n into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is tight if it "mentions" all coordinates. We study extremal properties of tight irreducible subcube partitions: minimal size, minimal weight, maximal number of points, maximal size, and maximal minimum dimension. We also consider the existence of homogeneous tight irreducible...

A subcube partition is a partition of the Boolean cube {0,1}^n into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is tight if it "mentions" all coordinates. We study extremal properties of tight irreducible subcube partitions: minimal size, minimal weight, maximal number of points, maximal size, and maximal minimum dimension. We also consider the existence of homogeneous tight irreducible...