The 3-design parameter formula for a lifted Reed–Muller code

Let $m\geq 3$. For the binary lifted Reed–Muller code ${\mathrm{RM}}_2(1,m)(2|2^2)$, let ${\mathbb{D}}_{3\times 2^{m-2}}({\mathrm{RM}}_2(1,m)(2|2^2))$ denote the design whose blocks are the supports of codewords of weight $3\times 2^{m-2}$. The design-parameter formula. The supports form a $3$-$(2^m,3\times 2^{m-2},\lambda_m)$ design, where

$$\lambda_m=\frac{2\binom{3\times 2^{m-2}}{3}(2^m-1)(2^m-2)}{3\binom{2^m}{3}}.$$

This gives the explicit parameter of the $3$-design arising from the weight-$3\times 2^{m-2}$ codewords in the lifted code; the surrounding result establishes the corresponding weight enumerator for all $m\geq 3$.