On intersections of Reed–Muller like codes

A binary code that has the parameters and possesses the main properties of the classical $r$ th-order Reed–Muller code $RM_{r,m}$ will be called an $r$ th-order Reed–Muller like code and will be denoted by $LRM_{r,m}$. The class of such codes contains the family of codes obtained by the Pulatov construction and also classical linear and $\mathbb{Z}_4$-linear Reed–Muller codes. We analyze the intersection problem for the Reed–Muller like codes. We prove that for any even $k$ in the interval $0\le k\le 2^{2\sum\limits_{i=0}^{r-1}\binom{m-1}{i}}$ there exist $LRM_{r,m}$ codes of order $r$ and length $2^m$ having intersection size $k$. We also prove that there exist two Reed–Muller like codes of order $r$ and length $2^m$ whose intersection size is $2k_1 k_2$ with $1\le k_s\le |RM_{r-1,m-1}|$, $s\in\{1,2\}$, for any admissible length starting from $16$.