The paired construction for Boolean functions on the slice

TL;DR

The paper determines the exact polynomial degree of paired Boolean functions on the Johnson slice $\binom{V}{k}$ by deriving an explicit formula for $\deg p^{(V,k)}_{I,J}$ in terms of $i=|I|$ and $j=|J|$, with a structured reduction from the middle-layer case $n=2k$ to the general case via derived, residual, and dual functions. It develops a comprehensive elementary framework avoiding spectral methods, tying degree to $t$-designs and the Johnson scheme's association structure, and proves monotonicity and case-by-case exact values. The results reveal that paired functions can achieve degrees strictly smaller than naive bounds, producing fixed-degree Boolean functions of small support and improving the minimal-size landscape beyond pencils in certain parameter regions, notably when $n=2k$ and $t$ is even and not in $\\{0,k\\}$. The work also connects to Hartman’s conjecture through degree arguments on designs and situates paired constructions as competitive alternatives to pencils for small-degree, small-size Boolean functions, with broader implications for design theory and combinatorial complexity.

Abstract

Let $V$ be a finite set of size $n$. We consider real functions on the "slice" $\binom{V}{k}$, which are also known as functions in the Johnson scheme. For $I \subseteq J \subseteq V$, the characteristic function of the set of all $K\in\binom{V}{k}$ with $I \subseteq K \subseteq J$ is called "basic". In this article, we investigate a construction arising as the sum of two "opposite" basic functions. In essentially all cases, these "paired" functions are Boolean. Our main result is the determination of the exact degree -- regarding a representation by an $n$-variable polynomial -- of all paired functions. The proof is elementary and does not involve any spectral methods. First, we settle the middle layer case $n=2k$ by identifying and combining various relations among the degrees involved. Then the general case is reduced to the middle layer situation by means of derived, reduced, and dual functions. Remarkably, in certain situations, the degree is strictly smaller than what is guaranteed by the elementary upper bound for the sum of functions. This makes paired functions good candidates for fixed-degree Boolean functions of small support size. As it turns out, for $n = 2k$ and even degree $t \notin \{0,k\}$, paired functions provide the smallest known non-zero Boolean functions, surpassing the $t$-pencils, which is the smallest known construction in all other cases.

Theorems & Definitions (82)

• Definition 1.4
• Lemma 1.5: Elementary bound
• Proof 1
• Theorem 1
• Lemma 2.8: LP-bound for the degree
• Lemma 3.1
• Proof 2
• Lemma 3.2: Pascal decomposition
• Proof 3
• Remark 3.3