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Hadamard Matrix of Order 668 Found: What It Means for the Conjecture

Hadamard Matrix of Order 668 Found: What It Means for the Conjecture — Hadamard

A Hadamard matrix of order 668 now exists. On 12 August 2026, Levent Alpöge, Philippe Voinov and Saul Reynolds-Haertle, working with Claude, announced one, closing what had been the smallest open case of the Hadamard conjecture for about 20 years. The conjecture is still unproven: according to an independent survey posted to arXiv on 19 October 2026, 95 orders up to 10,000 still have no known Hadamard matrix, and the smallest of them is 2396.

This article explains what was found, how it was checked, why the AI credit is still provisional, and why the matrix behind this blog’s name also sits at the centre of quantum computing.

What is a Hadamard matrix?

A Hadamard matrix is a square matrix with every entry equal to +1 or −1, in which any two different rows are orthogonal. Orthogonal means that if you multiply two rows entry by entry and add the results, you get zero. For an n×n matrix H this is written compactly as H·Hᵀ = nI, where I is the identity matrix (MathWorld).

The smallest interesting example has order 2, with rows (1, 1) and (1, −1). Jacques Hadamard studied these matrices in 1893, in work on the largest possible determinant of a matrix with bounded entries. For an n×n matrix with entries of absolute value at most 1, the determinant cannot exceed n^(n/2) in absolute value, and Hadamard matrices reach that bound exactly (Wikipedia: Hadamard matrix).

His work also showed a hard limit: a Hadamard matrix can exist only if its order is 1, 2 or a multiple of 4. He gave explicit examples of orders 12 and 20 (Ó Catháin, Monash seminar slides). The paper appeared in the Bulletin des Sciences Mathématiques. For more on the man himself, see our biography of Jacques Hadamard, from the prime number theorem to the H gate.

Where does the Hadamard conjecture come from?

The Hadamard conjecture says the necessary condition is also sufficient: for every multiple of 4, a Hadamard matrix of that order exists. Over 150 years, the conjecture has been built up through a series of constructions, each covering an infinite family of orders without ever covering all of them.

  • Sylvester (1867). Repeated doubling turns a matrix of order n into one of order 2n, so all powers of 2 are covered. The order-2^n matrix is the n-fold Kronecker product H₂^⊗n, and its entry in row a, column b is (−1)^(a·b), where a·b is the bitwise dot product of the two indices (Wikipedia: Hadamard transform).
  • Scarpis (1898). An early construction based on primes, which came before Paley’s more general one (“Paley and the Paley graphs”).
  • Paley (1933). A construction using finite fields. It gives order q+1 when q is a prime power with q ≡ 3 (mod 4), and order 2(q+1) when q ≡ 1 (mod 4) (Wikipedia: Paley construction).

The conjecture is often credited to Paley, but sources disagree about who should get the credit. Whoever stated it first, the open question has stayed the same: every known method leaves gaps, and each gap has to be filled one order at a time.

Why was 668 the smallest unknown Hadamard matrix order?

The simple arithmetic shows why 668 was hard. 668 = 4 × 167. Because 167 is odd and 334 is not a multiple of 4, the order cannot be reached by doubling a smaller Hadamard matrix, and it is not a product of two smaller valid orders. Paley’s construction does not apply either: 667 = 23 × 29 and 333 = 3² × 37 are not prime powers.

In 2000, the multiples of 4 up to 1000 with no known Hadamard matrix were 428, 668, 716, 764 and 892 (OEIS A007299). Two of them fell over the next decade:

Year Order What happened
2004–2005 428 Kharaghani and Tayfeh-Rezaie found it using Turyn-type sequences of lengths 36, 36, 36 and 35. IPM announced it in June 2004 and said a cluster of 16 PCs found it in 12 hours. The paper followed in 2005.
2007–2008 764 Đoković built two Goethals–Seidel-type matrices. The result was posted to arXiv in 2007 and published in Combinatorica 28(4), 2008.
2025 668 (partial) Eliahou built a “64-modular” Hadamard matrix of order 668, improving on a 32-modular one known since 2001. That is a step toward the goal, not a true Hadamard matrix.
2026 668 and 11 others Alpöge, Voinov and Reynolds-Haertle announced matrices for all twelve open orders below 2000.

Sources: Tayfeh-Rezaie’s announcement page, the IPM bulletin, Đoković on arXiv and Eliahou in the Australasian Journal of Combinatorics.

After 428 was settled, 668 became the smallest unknown Hadamard matrix order. In his 2025 paper, Eliahou still listed 668, 716 and 892 as the open orders up to 1000.

How the Hadamard matrix of order 668 was found

The August 2026 announcement covered much more than one order. It gave matrices for all twelve orders below 2000 that had no known construction: 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948 and 1964 (arXiv 2610.27175).

The team released the matrices but not the method used to build them. They have not published a peer-reviewed paper. Wikipedia’s entry on Alpöge notes that the result came outside formal peer review and was checked independently in the days that followed.

Publishing without a method is unusual, but in this case it does not weaken the result. A Hadamard matrix certifies itself. Anyone can check one without trusting how it was made.

How was the matrix verified?

Checking takes one matrix multiplication: compute H·Hᵀ and confirm it equals 668·I. That takes seconds on a laptop, which explains why the first independent checks came within days.

The fuller analysis came on 19 October 2026, when Egan, Kotsireas, London and Ó Catháin posted “Recent progress in the construction of Hadamard matrices”. They verified all twelve matrices and also worked out their structure:

  • Each matrix is a Goethals–Seidel array over an abelian group. This is the same broad family Đoković used for order 764.
  • Nine are built over cyclic groups, and four of those nine include a border of width 4.
  • Three are built over non-cyclic abelian groups.
  • The order-668 matrix comes from Legendre quadruples.

The survey adds results of its own. The authors constructed Hadamard matrices for 16 more orders that were open in the published literature. The paper therefore does two things: it confirms the announced matrices, and it extends the known range of orders.

Did AI solve the Hadamard matrix problem?

Epoch AI tracks open mathematical problems as part of its FrontierMath project. On its page for the Hadamard Matrix of Order 668, dated 2026-08-12, it marks the problem as solved by AI and credits the three researchers together with Claude.

Epoch calls the label provisional. According to its open-problems changelog, the classification will be revised if the team’s later reports show that humans supplied the core ideas. Until the construction method is published, nobody outside the team can say how the work was divided between the researchers and the model.

The facts that can be stated now are these. The matrices are correct, independent mathematicians have explained their structure, and the question of credit is open because the method has not been released.

What is still open after 668?

The Hadamard conjecture is still unproven. The 19 October survey counts 95 orders up to 10,000 with no known Hadamard matrix. The smallest is now 2396 = 4 × 599. Separately, Schneider constructed order 2060 (arXiv 2610.27175; MathWorld).

Filling in individual orders, however many, cannot prove the conjecture. A proof would need a construction or existence argument that works for every multiple of 4, and no such argument is known. The 2026 results moved the smallest open order from 668 to 2396, but there are infinitely many cases and the general question is exactly where it was.

Why Hadamard matrices matter for quantum computing and coding

The 2×2 Sylvester matrix, scaled by 1/√2, is the Hadamard gate H. It takes a qubit in state |0⟩ or |1⟩ to an equal superposition of both. Applying H to each of n qubits gives H^⊗n, also called the Walsh–Hadamard transform. Apart from the factor 1/√(2^n), it is exactly Sylvester’s matrix of order 2^n (Wikipedia: Hadamard transform; Intro to Quantum Computing for Statisticians, arXiv 2112.06587).

This layer of Hadamard gates is the first step of many quantum algorithms, including the search algorithm covered in our explainer on what Grover’s and BHT’s algorithms really do to SHA-256. Quantum hardware uses only the power-of-two orders, which Sylvester settled in 1867. The orders at the open frontier, such as 668 and 2396, are a purely combinatorial question.

Hadamard matrices have also been used in engineering for a long time. Mariner 9, launched in 1971, encoded its images with a [32, 6, 16] Hadamard code, which is the first-order Reed–Muller code. Its codewords came from a 32×32 Hadamard matrix stacked with its negative. That allowed 64 grey levels per pixel and the correction of up to 7 bit errors per codeword. The probe returned 7,329 images covering 85% of Mars (Wikipedia: Hadamard code; John D. Cook). These figures come from secondary sources, not a primary NASA document.

This blog is published by Hadamard Corp, which builds BB84, a hash-based Ethereum Layer 2 designed to avoid the ECDSA signatures that a large quantum computer could break. For background on that threat, see whether quantum computers can break Ethereum.

FAQ

Does finding a 668×668 Hadamard matrix prove the Hadamard conjecture?

No. It settles specific orders: 668 and eleven others below 2000. The conjecture covers every multiple of 4, and 95 orders up to 10,000 are still open, starting with 2396.

How do you check that a matrix is really a Hadamard matrix?

Confirm that every entry is ±1 and that H·Hᵀ = nI. For n = 668 this takes seconds on an ordinary laptop, which is why independent checks came so quickly.

Did AI really solve the problem?

Epoch AI lists it as solved by AI, crediting three researchers and Claude, but says the label is provisional. The team has not yet published how the matrices were built, so the split between human and model contributions is not yet known.

What does a Hadamard matrix have to do with the Hadamard gate?

The Hadamard gate is the 2×2 Sylvester matrix scaled by 1/√2. Applying it to n qubits gives H^⊗n, which is the normalised Sylvester Hadamard matrix of order 2^n.

Sources