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Hadamard Difference Set

A Hadamard difference set is a difference set with parameters (4u^2,2u^2-u,u^2-u) for a positive integer u. The incidence matrix construction produces a regular Hadamard matrix of order 4u^2 (Beth et al. 1999). Indeed, if A is the incidence matrix of the translates of the difference set, then H=J-2A satisfies HH^T=4u^2I, where J is the matrix with every entry equal to 1 and I is the identity matrix. Every row and column of H sums to 2u, which is the regularity condition.

Schmidt (2026) proved that no group of group order 4p^2 contains such a set when u=p>=7 is prime and p=3 (mod4). Schmidt (2026) reports that AI generated the proof for the remaining group case and that the author checked it.

See also

Difference Set, Hadamard Matrix

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References

Beth, T.; Jungnickel, D.; and Lenz, H. Design Theory, 2nd ed. Cambridge, England: Cambridge University Press, 1999.Schmidt, B. "Nonexistence of Hadamard Difference Sets in Groups of Order 4p^2 with p=3 (mod4), p>=7." 27 Oct 2026. https://arxiv.org/abs/2610.37417.

Cite this as:

Weisstein, Eric W. "Hadamard Difference Set." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HadamardDifferenceSet.html

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