Preprint · August 2026

Two H-inequivalent Hadamard matrices of order 668

Combinatorics (05B20, 05B10) · 4 pages · exact integer arithmetic throughout

Abstract

Four explicit subsets of ℤ₁₆₆ form a cyclic difference family and define a bordered Goethals–Seidel Hadamard matrix H of order 668. The four border rows and columns form a Hall set. A paired Hall switch therefore gives another Hadamard matrix H^★. Exact enumeration shows that the Hall row and column sets are unique, and an intrinsic four-row correlation profile distinguishes the two matrices: its maximum is 148 for H and 164 for H^★. Thus H and H^★ are not H-equivalent, even if transposition is admitted.

At a glance

Order	668
Difference family	(166; 82, 83, 83, 83; 164)
Construction	Bordered Goethals–Seidel array over four circulants
Switch	Paired Hall switch on P = {1,2,3,4}; 1328 entries change
Invariant	Four-row correlation profile Φ_M, over 194,289,056 terms
Separation	max supp Φ_H = 148 vs. max supp Φ_H★ = 164
All 16 switch masks	Hadamard; fall into exactly two H-classes

Matrix hashes

H	19f435b31eb6561dd97356b761e9b0f174824e42c215c33fbafc20f9b1b20744
H^★	08428eb6779986c50ee2997584daa5c87e00520cc1e6f62d7793dc519f85ea45

Row-major bytes, encoding +1 by 1 and −1 by 0.

Full text

References

L. Alpöge, X post announcing Hadamard constructions including order 668, 12 August 2026. x.com/__alpoge__
J.-M. Goethals and J. J. Seidel, Orthogonal matrices with zero diagonal, Canad. J. Math. 19 (1967), 1001–1010.
B. D. McKay, Hadamard equivalence via graph isomorphism, Discrete Math. 27 (1979), no. 2, 213–214.
W. P. Orrick, Switching operations for Hadamard matrices, SIAM J. Discrete Math. 22 (2008), no. 1, 31–50.