Equitable [[2,10],[6,6]]-partitions of the 12-cube

TL;DR

This paper classifies equitable partitions of the 12-cube with a specific quotient matrix, revealing 103 equivalence classes, and explores related orthogonal arrays and correlation-immune Boolean functions using computer-aided methods.

Contribution

It provides the first complete classification of certain equitable partitions and associated orthogonal arrays in the 12-cube, including the enumeration of equivalence classes and analysis of their properties.

Findings

103 equivalence classes of the considered objects

Only two almost-OA(1536,12,2,8) found

40 classes of pairs of disjoint simple OA(1536,12,2,7)

Abstract

We describe the computer-aided classification of equitable partitions of the $12$-cube with quotient matrix $[[2,10],[6,6]]$, or, equivalently, simple orthogonal arrays OA$(1536,12,2,7)$, or order-$7$ correlation-immune Boolean functions in $12$ variables with $1536$ ones (which completes the classification of unbalanced order-$7$ correlation-immune Boolean functions in $12$ variables). We find that there are $103$ equivalence classes of the considered objects, and there are only two almost-OA$(1536,12,2,8)$ among them. Additionally, we find that there are $40$ equivalence classes of pairs of disjoint simple OA$(1536,12,2,7)$ (equivalently, equitable partitions of the $12$-cube with quotient matrix $[[2,6,4], [6,2,4], [6,6,0]]$) and discuss the existence of a non-simple OA$(1536,12,2,7)$. Keywords: orthogonal arrays, correlation-immune Boolean functions, equitable partitions, perfect colorings, intriguing sets.