[Paper Review] Inequivalence of skew Hadamard difference sets and triple intersection numbers modulo a prime

Koji Momihara|arXiv (Cornell University)|Aug 6, 2013

graph theory and CDMA systems7 references3 citations

TL;DR

This paper introduces a new invariant—triple intersection numbers modulo a prime—to prove that infinitely many skew Hadamard difference sets constructed by Feng and Xiang are inequivalent to the classical Paley difference sets. By leveraging recursive lifting properties and character sum analysis, the authors establish that these Feng-Xiang constructions yield distinct combinatorial objects even when parameters are similar.

ABSTRACT

Recently, Feng and Xiang \cite{FX113} found a new construction of skew Hadamard difference sets in elementary abelian groups. In this paper, we introduce a new invariant for equivalence of skew Hadamard difference sets, namely triple intersection numbers modulo a prime, and discuss inequivalence between Feng-Xiang skew Hadamard difference sets and the Paley difference sets. As a consequence, we show that their construction produces infinitely many skew Hadamard difference sets inequivalent to the Paley difference sets.

Motivation & Objective

• Address the long-standing open problem of whether skew Hadamard difference sets other than Paley's exist in abelian groups.
• Provide a theoretical framework to prove inequivalence without relying on computationally intensive methods.
• Overcome the limitation of standard invariants (e.g., p-ranks) that are determined solely by parameters and thus fail to distinguish inequivalent skew Hadamard difference sets.
• Establish a new theoretical tool—triple intersection numbers modulo a prime—for detecting inequivalence in combinatorial designs over finite fields.
• Demonstrate that the Feng-Xiang construction generates infinitely many skew Hadamard difference sets not equivalent to the Paley difference sets.

Proposed method

• Define a new invariant: triple intersection numbers modulo a prime, derived from character sum properties over finite fields.
• Use the Davenport-Hasse product formula to analyze Gauss sums and derive divisibility conditions for character sums.
• Leverage the lifting property of Feng-Xiang difference sets: if a set is skew Hadamard in $\mathbb{F}_q$, its lift to $\mathbb{F}_{q^t}$ remains skew Hadamard when $\gcd(t,N)=1$.
• Apply recursive techniques to compare triple intersection numbers across lifts, showing that these numbers differ between Feng-Xiang and Paley difference sets.
• Utilize cyclotomic classes of order $N = 2p_1^m$ in $\mathbb{F}_q$, where $q = p^{fs}$, with $p \equiv 3 \pmod{4}$ and $f = \mathrm{ord}_N(p) = \phi(N)/2$.
• Establish that the triple intersection numbers modulo a prime are preserved under lifting when $\gcd(t,N)=1$, enabling recursive comparison across field extensions.

Experimental results

Research questions

• RQ1 Are there infinitely many skew Hadamard difference sets in abelian groups that are inequivalent to the Paley difference sets?
• RQ2 Can a theoretical invariant be constructed to distinguish skew Hadamard difference sets beyond parameter-based invariants?
• RQ3 Do the Feng-Xiang skew Hadamard difference sets, which are closed under lifting, produce distinct combinatorial objects compared to the Paley construction?
• RQ4 Can the triple intersection numbers modulo a prime serve as a reliable invariant for detecting inequivalence in skew Hadamard difference sets?
• RQ5 Are there other constructions of skew Hadamard difference sets with the lifting property beyond the Paley and Feng-Xiang families?

Key findings

• The paper proves that the Feng-Xiang skew Hadamard difference sets are inequivalent to the Paley difference sets using the new invariant of triple intersection numbers modulo a prime.
• It is shown that for the parameter set $(v,k,\lambda) = (67,3,14)$, the triple intersection numbers $n_t$ are at least 3 for all $t$ in the Feng-Xiang constructions, while for the Paley set, $n_t = 2$ for all $t$.
• For $(v,k,\lambda) = (107,3,14)$, the triple intersection numbers $n_3 = 1$ for one Feng-Xiang set, while $n_t = 2$ for all $t$ in the Paley set, proving inequivalence.
• The construction yields infinitely many skew Hadamard difference sets inequivalent to the Paley difference sets, as the lifting process preserves the invariant and allows recursive distinction.
• The method successfully detects inequivalence without relying on computer search, providing a theoretical alternative to computational verification.
• The paper generalizes the Feng-Xiang construction to a broader class of parameters via Theorem 13, where $2 \in \langle p \rangle \pmod{p_1^m}$ and $f = \mathrm{ord}_N(p)$ is odd, extending the range of applicable cases.