article published yes Vivian Zieve Kuperberg author c3bac823-112d-11f0-a3f5-c264f852e697 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. Wiley2016 eng modular hadamard matricesmodular symmetric designs 1063-8539 1520-6610 1409.014810.1002/jcd.21522 249393-405 yes 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> V.Z. Kuperberg, Journal of Combinatorial Designs 24 (2016) 393–405. 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> 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>. Kuperberg VZ. 2016. Hadamard matrices modulo p and small modular Hadamard matrices. Journal of Combinatorial Designs. 24(9), 393–405. 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. 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>. 222022026-06-29T13:00:27Z2026-07-14T11:35:58Z