---
OA_place: repository
OA_type: green
_id: '22202'
abstract:
- lang: eng
  text: We use modular symmetric designs to study the existence of Hadamard matrices
    modulo certain primes. We solve the 7-modular and 11-modular versions of the Hadamard
    conjecture for all but a finite number of cases. In doing so, we state a conjectural
    sufficient condition for the existence of a p-modular Hadamard matrix for all but
    finitely many cases. When 2 is a primitive root of a prime p, we conditionally
    solve this conjecture and therefore the p-modular version of the Hadamard conjecture
    for all but finitely many cases when p ≡ 3(mod 4), and prove a weaker result for
    p ≡ 1 (mod 4). Finally, we look at constraints on the existence of m-modular Hadamard
    matrices when the size of the matrix is small compared to m.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Vivian Zieve
  full_name: Kuperberg, Vivian Zieve
  id: c3bac823-112d-11f0-a3f5-c264f852e697
  last_name: Kuperberg
citation:
  ama: Kuperberg VZ. Hadamard matrices modulo p and small modular Hadamard matrices.
    <i>Journal of Combinatorial Designs</i>. 2016;24(9):393-405. doi:<a href="https://doi.org/10.1002/jcd.21522">10.1002/jcd.21522</a>
  apa: Kuperberg, V. Z. (2016). Hadamard matrices modulo p and small modular Hadamard
    matrices. <i>Journal of Combinatorial Designs</i>. Wiley. <a href="https://doi.org/10.1002/jcd.21522">https://doi.org/10.1002/jcd.21522</a>
  chicago: Kuperberg, Vivian Zieve. “Hadamard Matrices modulo p and Small Modular
    Hadamard Matrices.” <i>Journal of Combinatorial Designs</i>. Wiley, 2016. <a href="https://doi.org/10.1002/jcd.21522">https://doi.org/10.1002/jcd.21522</a>.
  ieee: V. Z. Kuperberg, “Hadamard matrices modulo p and small modular Hadamard matrices,”
    <i>Journal of Combinatorial Designs</i>, vol. 24, no. 9. Wiley, pp. 393–405, 2016.
  ista: Kuperberg VZ. 2016. Hadamard matrices modulo p and small modular Hadamard
    matrices. Journal of Combinatorial Designs. 24(9), 393–405.
  mla: Kuperberg, Vivian Zieve. “Hadamard Matrices modulo p and Small Modular Hadamard
    Matrices.” <i>Journal of Combinatorial Designs</i>, vol. 24, no. 9, Wiley, 2016,
    pp. 393–405, doi:<a href="https://doi.org/10.1002/jcd.21522">10.1002/jcd.21522</a>.
  short: V.Z. Kuperberg, Journal of Combinatorial Designs 24 (2016) 393–405.
date_created: 2026-06-29T13:00:27Z
date_published: 2016-09-01T00:00:00Z
date_updated: 2026-07-14T11:35:58Z
day: '01'
doi: 10.1002/jcd.21522
extern: '1'
external_id:
  arxiv:
  - '1409.0148'
intvolume: '        24'
issue: '9'
keyword:
- modular hadamard matrices
- modular symmetric designs
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1409.0148
month: '09'
oa: 1
oa_version: Preprint
page: 393-405
publication: Journal of Combinatorial Designs
publication_identifier:
  eissn:
  - 1520-6610
  issn:
  - 1063-8539
publication_status: published
publisher: Wiley
quality_controlled: '1'
scopus_import: '1'
status: public
title: Hadamard matrices modulo p and small modular Hadamard matrices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 24
year: '2016'
...