Classification of Δ-Divisible Linear Codes Spanned by Codewords of Weight Δ

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.