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An Explicit Skew-Hadamard Matrix of Order 1252 via Cyclotomic Unions

Amira Karoui

Abstract

We construct a skew-Hadamard matrix of order 1252 = 2(5^4 + 1) using a bordered skew-Hadamard difference family over GF(5^4), with blocks given as unions of cyclotomic classes of order N = 16. This order has been reported as missing in some widely used open-source computational tables; we provide an explicit instance together with verification artifacts. We prove the structural prerequisites for the bordered construction (skew-symmetry of one block and the constant autocorrelation-sum condition), and we compute algebraic invariants to facilitate classification: the associated tournament adjacency matrix has full rank over GF(2), and the matrix has full rank over GF(3) and GF(5). We also exhibit an explicit affine subgroup of the automorphism group of size 24 375. All claims are supported by a reproducible artifact bundle including the explicit matrix and verification logs.

An Explicit Skew-Hadamard Matrix of Order 1252 via Cyclotomic Unions

Abstract

We construct a skew-Hadamard matrix of order 1252 = 2(5^4 + 1) using a bordered skew-Hadamard difference family over GF(5^4), with blocks given as unions of cyclotomic classes of order N = 16. This order has been reported as missing in some widely used open-source computational tables; we provide an explicit instance together with verification artifacts. We prove the structural prerequisites for the bordered construction (skew-symmetry of one block and the constant autocorrelation-sum condition), and we compute algebraic invariants to facilitate classification: the associated tournament adjacency matrix has full rank over GF(2), and the matrix has full rank over GF(3) and GF(5). We also exhibit an explicit affine subgroup of the automorphism group of size 24 375. All claims are supported by a reproducible artifact bundle including the explicit matrix and verification logs.
Paper Structure (15 sections, 4 theorems, 8 equations, 1 table)

This paper contains 15 sections, 4 theorems, 8 equations, 1 table.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Scope.
  4. Preliminaries
  5. Finite fields and cyclotomic classes
  6. Autocorrelation for subsets of an abelian group
  7. Construction
  8. Blocks as cyclotomic unions in $GF(5^4)$
  9. Certification (artifact-based verification)
  10. Algebraic invariants
  11. $p$-rank fingerprints
  12. Automorphisms (explicit affine subgroup)
  13. Practical use case: Edge AI compression
  14. Conclusion
  15. Artifact bundle and reproducibility (brief)

Key Result

Lemma 1

Let $G$ be an abelian group of order $v$, and let $\{D_0,D_1\}$ satisfy the bordered SHDF condition (Definition def:shdf) in $G$. Then the standard bordered group-developed (Goethals--Seidel type) array associated to $\{D_0,D_1\}$ yields a skew-Hadamard matrix of order $2(v+1)$.

Theorems & Definitions (9)

  • Definition 1: Bordered SHDF condition
  • Lemma 1: Bordered construction (Goethals--Seidel type), cf. momihara2018handbookCD
  • Remark 1
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • Remark 2