# On $(2n/3-1)$-resilient $(n,2)$-functions

## TL;DR

This paper investigates the structure and construction of $(2n/3-1)$-resilient $(n,2)$-functions, exploring their connections with equitable partitions, Latin hypercubes, and perfect codes, and characterizing their properties.

## Contribution

It introduces new constructions of resilient functions, links them to combinatorial objects, and characterizes non-full-rank and reducible functions within this class.

## Key findings

- Constructed new $(2n/3-1)$-resilient functions and partitions.
- Established connections with Latin hypercubes and perfect codes.
- Characterized non-full-rank and reducible functions.

## Abstract

A $\{00,01,10,11\}$-valued function on the vertices of the $n$-cube is called a $t$-resilient $(n,2)$-function if it has the same number of $00$s, $01$s, $10$s and $11$s among the vertices of every subcube of dimension $t$. The Friedman and Fon-Der-Flaass bounds on the correlation immunity order say that such a function must satisfy $t\le 2n/3-1$; moreover, the $(2n/3-1)$-resilient $(n,2)$-functions correspond to the equitable partitions of the $n$-cube with the quotient matrix $[[0,r,r,r],[r,0,r,r],[r,r,0,r],[r,r,r,0]]$, $r=n/3$. We suggest constructions of such functions and corresponding partitions, show connections with Latin hypercubes and binary $1$-perfect codes, characterize the non-full-rank and the reducible functions from the considered class, and discuss the possibility to make a complete characterization of the class.