[Paper Review] Supplementary difference sets with symmetry for Hadamard matrices

Dragomir Ž. Djoković|arXiv (Cornell University)|Mar 29, 2009

graph theory and CDMA systems9 references3 citations

TL;DR

This paper presents new supplementary difference sets (SDSs) with symmetric or skew properties in finite abelian groups, enabling the construction of previously unknown skew Hadamard matrices of order 188, 244, and 508 via the Goethals–Seidel array. It introduces five new multicirculant Williamson matrices over elementary abelian groups of order 25, 27, and 49, and constructs a new $(127,57,76)$ difference family yielding a skew Hadamard matrix of order 508 and a BIBD with the same parameters.

ABSTRACT

First we give an overview of the known supplementary difference sets (SDS) (A_i), i=1..4, with parameters (n;k_i;d), where k_i=|A_i| and each A_i is either symmetric or skew and k_1 + ... + k_4 = n + d. Five new Williamson matrices over the elementary abelian groups of order 25, 27 and 49 are constructed. New examples of skew Hadamard matrices of order 4n for n=47,61,127 are presented. The last of these is obtained from a (127,57,76)-difference family that we have constructed. An old non-published example of G-matrices of order 37 is also included.

Motivation & Objective

• To extend the construction of skew Hadamard matrices beyond known circulant Williamson matrices by exploring supplementary difference sets (SDSs) with symmetry types (ksss), (kkss), and (kkks).
• To construct new multicirculant Williamson matrices of order 49 over the elementary abelian group $F_{49}$, providing non-equivalent examples to Wilson's construction.
• To generate a new $(127,57,76)$ difference family, yielding a skew Hadamard matrix of order 508 and a balanced incomplete block design (BIBD).
• To recover and present an old unpublished example of $G$-matrices of order 37, contributing to the known list of skew-type Hadamard matrices.

Proposed method

• The Goethals–Seidel array is used to construct Hadamard matrices of order $4n$ from quadruples of $n\times n$ binary matrices derived from SDSs in a finite abelian group $\mathcal{A}$ of order $n$.
• Supplementary difference sets $(A_1, A_2, A_3, A_4)$ are defined with parameters $(n; k_1,k_2,k_3,k_4; \lambda)$ satisfying $\lambda = \sum k_i - n$, ensuring the resulting matrix is Hadamard.
• Each $A_i$ is required to be either symmetric ($A_i = -A_i$) or skew ($\mathcal{A} = A_i \cup -A_i \cup \{0\}$), and the symmetry type (e.g., (ksss)) is used to classify the SDSs.
• Binary matrices $A_i^c$ are constructed from characteristic functions of $A_i$, and their properties ensure the Goethals–Seidel array yields a Hadamard matrix.
• For prime $n=127$, the construction uses cosets of the subgroup $H=\{1,2,4,8,16,32,64\}$ in $\mathbb{Z}_{127}^*$, with index sets defining three symmetric blocks.
• The method combines a classical Paley skew difference set with a new $(127,57,76)$ difference family to produce a new SDS of type (ks**), leading to a skew Hadamard matrix of order 508.

Experimental results

Research questions

• RQ1 Can new supplementary difference sets with symmetry types (ksss), (kkss), or (kkks) be constructed to yield previously unknown skew Hadamard matrices?
• RQ2 Are there non-cyclic, non-equivalent SDSs of type (ssss) in elementary abelian groups of order $5^2$, $3^3$, and $7^2$ that give rise to multicirculant Williamson matrices?
• RQ3 Can a $(127,57,76)$ difference family be constructed to generate a skew Hadamard matrix of order 508?
• RQ4 What is the role of coset structures in $\mathbb{Z}_{127}^*$ in enabling the construction of symmetric SDSs with high symmetry and optimal parameters?
• RQ5 How do the symmetry types of SDSs influence the structure and existence of associated skew Hadamard matrices?

Key findings

• Five new multicirculant Williamson matrices were constructed: two for order 25, two for order 27, and one for order 49, all over elementary abelian groups.
• A new skew Hadamard matrix of order 188 was constructed from an SDS with parameters $(47;23,21,19,19;35)$ and type (ks**), where the first block is the classical skew difference set of squares in $\mathbb{Z}_{47}$.
• A new skew Hadamard matrix of order 244 was constructed from a cyclic SDS with parameters $(61;30,28,27,24;48)$ and type (k**s), confirming the existence of such matrices for $n=61$.
• A new skew Hadamard matrix of order 508 was constructed from a $(127,57,76)$ difference family, which also yields a balanced incomplete block design (BIBD) with the same parameters.
• The new SDS for $n=127$ uses cosets of the subgroup $H=\{1,2,4,8,16,32,64\}$ in $\mathbb{Z}_{127}^*$, with index sets $J_1, J_2, J_3$ defining three symmetric blocks, and is not equivalent to previous constructions.
• An old unpublished example of $G$-matrices of order 37 was recovered and included, contributing to the known list of skew-type Hadamard matrices.