Classification of Δ-Divisible Linear Codes Spanned by Codewords of Weight Δ
Michael Kiermaier, Sascha Kurz · IEEE Transactions on Information Theory · 2023
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.