Classification problem for order-six complex Hadamard matrices

Classify, up to standard equivalence, all matrices $H\in\mathbb{C}^{6\times 6}$ satisfying $|H_{ij}|=1$ for every $i,j$ and $HH^{*}=6I_{6}$ . The conjecture asserts that every such matrix is standard-equivalent to either Tao's isolated matrix, a member of Karlsson's three-parameter family, or a member of the four-parameter family $G_{6}^{(4)}$ , where standard equivalence is generated by permuting rows or columns and multiplying rows or columns by complex phases.

Progress summary

Solved

A new manuscript claims to finish the six-by-six classification, but the claim has not yet been independently checked.

The problem seeks an exhaustive classification of complex Hadamard matrices of order $6$ . A 2010 conjecture proposed that Tao’s isolated matrix together with the known continuous families gives the complete list.

Known results

In $2007$ , only a restricted symmetric, real-diagonal subclass was completely classified (Matolcsi and Szöllősi).
In $2010$ , every $H_{2}$ -reducible matrix was shown to lie in Karlsson’s three-parameter family (Karlsson).
In $2010$ , the four-parameter family $G_{6}^{(4)}$ was constructed, and its exhaustiveness with Karlsson’s family and Tao’s matrix was conjectured.
In $2014$ , the full classification was still explicitly described as open.

August 2026 claimed classification

On August 18, 2026, Francesco Szöllősi and Nico Stollenwerk claimed a proof of Szöllősi’s conjecture and recovery of all classes outside Tao’s isolated matrix and Karlsson’s family. The manuscript is unrefereed, and no independent verification or objection report was found.

Current status (as of August 2026): A complete classification is claimed in an unrefereed manuscript, but its proof remains unverified.

Sources

Sources & referencesView supporting material

Primary source

arXiv

Additional references

A Complete Classification of Complex Hadamard Matrices of Order Six — arXiv — Francesco Szöllősi, Nico Stollenwerk

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