On Intersections of Reed–Muller Like Codes
F. I. Solov’eva · Problems of Information Transmission · 2021
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.