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A class of skew-regular quaternary Hadamard matrices

Hadi Kharaghani, Vlad Zaitsev

TL;DR

This work generalizes the notion of skew-regular Hadamard matrices to the quaternary case and provides an explicit construction of skew-regular quaternary Hadamard matrices of order $1+p^2$ for every odd prime power $p$, achieving row sums $1-pi$ and enabling the generation of infinite families via a recursive COD-based framework. By exploiting the skew-core and a COD-driven recursion, the authors obtain larger Hadamard matrices with controlled row-sum patterns, including skew-semi-regular variants, and demonstrate substantial applications to the excess problem, producing matrices of order $4+4p^2$ with maximum excess $8p(1+p^2)$. The approach blends finite-field character sums, conference matrices, and quaternary orthogonal designs to yield both theoretical and practical extensions of Hadamard matrix families. Overall, the paper contributes new constructions, recursive methods, and excess-optimal instances that advance the theory and applications of quaternary Hadamard matrices.

Abstract

We construct a class of skew-regular quaternary Hadamard matrices of order $1+p^2$ for every odd prime power $p$. These matrices possess a row sum of $1-pi$. Applications include the generation of Hadamard matrices of order $4(1+p^2)$ with an excess of $8p(1+p^2)$.

A class of skew-regular quaternary Hadamard matrices

TL;DR

This work generalizes the notion of skew-regular Hadamard matrices to the quaternary case and provides an explicit construction of skew-regular quaternary Hadamard matrices of order for every odd prime power , achieving row sums and enabling the generation of infinite families via a recursive COD-based framework. By exploiting the skew-core and a COD-driven recursion, the authors obtain larger Hadamard matrices with controlled row-sum patterns, including skew-semi-regular variants, and demonstrate substantial applications to the excess problem, producing matrices of order with maximum excess . The approach blends finite-field character sums, conference matrices, and quaternary orthogonal designs to yield both theoretical and practical extensions of Hadamard matrix families. Overall, the paper contributes new constructions, recursive methods, and excess-optimal instances that advance the theory and applications of quaternary Hadamard matrices.

Abstract

We construct a class of skew-regular quaternary Hadamard matrices of order for every odd prime power . These matrices possess a row sum of . Applications include the generation of Hadamard matrices of order with an excess of .

Paper Structure

This paper contains 10 sections, 8 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A class of skew-regular matrices
  4. A recursive construction
  5. Quaternary orthogonal designs
  6. A recursive construction
  7. Skew semi-regular quaternary Hadamard matrices
  8. An application to the excess problem
  9. Example of main construction
  10. Skew-regular quaternary Hadamard of order 50 and equivalence to row sums $\pm 5 \pm 5i$.

Key Result

Lemma 1

Let $p$ be an odd prime and set $q=p^2$. For $t\in\mathbb{F}_{q}$ and define Then

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Definition 3
  • Lemma 3
  • proof
  • Definition 4
  • ...and 13 more