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

Denis S. Krotov

TL;DR

This work completes the classification of unbalanced order-$7$ correlation-immune Boolean functions in $n=12$ variables by computer-aided classification of equitable $[[2,10],[6,6]]$-partitions of the $12$-cube, equivalently simple OA$(1536,12,2,7)$. The authors develop a refined, hierarchical search using local partitions, a reconstruction step via exact-cover, and a subdivision into square, heavy, and square-free subfamilies to manage computation, ultimately counting $103$ equivalence classes and identifying $2$ that are almost-OA$(1536,12,2,7+)$ while also cataloging $40$ pairs of disjoint OA$(1536,12,2,7)$. They analyze cycle structures, automorphism groups, and Fourier spectra to characterize properties and connect to known constructions, and they discuss related structures such as splitting into equitable $3$-partitions and the existence of non-simple OA$(1536,12,2,7)$. The results advance the understanding of the landscape of correlation-immune Boolean functions and their OA-counterparts, with implications for combinatorial design and cryptographic applications.

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.

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

TL;DR

This work completes the classification of unbalanced order- correlation-immune Boolean functions in variables by computer-aided classification of equitable -partitions of the -cube, equivalently simple OA. The authors develop a refined, hierarchical search using local partitions, a reconstruction step via exact-cover, and a subdivision into square, heavy, and square-free subfamilies to manage computation, ultimately counting equivalence classes and identifying that are almost-OA while also cataloging pairs of disjoint OA. They analyze cycle structures, automorphism groups, and Fourier spectra to characterize properties and connect to known constructions, and they discuss related structures such as splitting into equitable -partitions and the existence of non-simple OA. The results advance the understanding of the landscape of correlation-immune Boolean functions and their OA-counterparts, with implications for combinatorial design and cryptographic applications.

Abstract

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

Paper Structure

This paper contains 17 sections, 5 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Classification
  3. Local partitions
  4. A reconstruction step
  5. How to reduce the amount of computation
  6. The square partitions
  7. The heavy partitions
  8. The square-free partitions
  9. Results and properties
  10. Cycle length
  11. Automorphisms
  12. Good orthogonal arrays and Fourier coefficients
  13. Known constructions
  14. Related structures
  15. Splitting into equitable $3$-partitions
  16. ...and 2 more sections

Key Result

Lemma 1

There are $66462606$$2$-local partitions $(C_+,C_-)$ with $\bar{0}$, $\bar{e}_{12}$, $\bar{e}_{11}$, $\bar{e}_{11}+\bar{e}_{12}\in C_+$, which are partitioned into $60$$2$-equivalence classes or $286$$(2,2)$-equivalence classes.

Theorems & Definitions (11)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Lemma 3
  • Remark 2
  • Lemma 4
  • proof
  • Theorem 1
  • ...and 1 more