Classification of $\Delta$-divisible linear codes spanned by codewords of weight $\Delta$

We classify all $q$-ary $\Delta$-divisible linear codes which are spanned by codewords of weight $\Delta$. 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 $\Delta=4$ of our classification. As an application, we give an alternative proof of a theorem of Liu on binary $\Delta$-divisible codes of length $4\Delta$ in the projective case.