The circulant Hadamard and Barker-sequence conjectures

1 · The essentials

This family concerns square arrangements of plus and minus signs with very strict rules. Each row repeats the previous row with a circular shift, and any two different rows agree in exactly as many places as they disagree. The main manuscript establishes that such arrangements exist only with one row or four rows. It is marked as Lean-formalized, meaning that the main result has a computer-checked mathematical formalization. A related question asks for lists of plus and minus signs that nearly cancel when compared with shifted copies of themselves, counting only the overlapping positions. Combining the matrix result with earlier results, the manuscript gives the complete list of possible lengths greater than one: $2, 3, 4, 5, 7, 11, 13$ .

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a row of reversible tiles, each showing either a plus or a minus, and copies of that row slid sideways. The sequence question asks whether matching and mismatching tiles in every nonzero overlap can balance to within one; the matrix question instead uses shifts that wrap around.

Why it matters

The result settles the circulant Hadamard conjecture by excluding every order except 1 and 4. Together with classical Barker-sequence results, it also yields the exact allowable lengths for Barker sequences longer than one. The connection lets a restriction on square matrices finish a classification of sequences.

2 · Going further

A real Hadamard matrix is a square matrix with entries $+ 1$ or $- 1$ whose distinct rows have inner product zero. It is circulant when its rows are successive circular shifts of one row. The precise main statement is that a real circulant Hadamard matrix has order $1$ or $4$ ; the repository summary states the existence classification as exactly these two orders.

For a binary sequence $a_{1}, \dots, a_{n}$ with entries in ${+ 1, - 1}$ , the nontrivial aperiodic autocorrelation at shift $k$ is $C_{k} = \sum_{i = 1}^{n - k} a_{i} a_{i + k}$ , for $1 \leq k < n$ . The Barker condition is $∣ C_{k} ∣ \leq 1$ at every such shift. The manuscript states that, for $n > 1$ , Barker sequences exist exactly when $n \in {2, 3, 4, 5, 7, 11, 13}$ .

The supplied abstract announces the matrix result and its Barker-sequence consequence but does not describe the argument establishing the matrix restriction. The repository summary supplies the dependency for the consequence: the new matrix result is combined with classical Barker-sequence results. Only one manuscript is supplied, so no companion-manuscript arguments or further relationships can be described from this source.

Useful notions

Square matrices with plus and minus entries
Circular shifts
Inner products and cancellation
Comparing a sequence with a shifted copy

What was known before / what is new

Before

The source identifies the circulant Hadamard and Barker-sequence statements as conjectures. It also refers to classical Barker-sequence results used in obtaining the final length classification. It does not specify those earlier results or their individual contributions.

New

The main manuscript establishes the restriction of real circulant Hadamard matrices to orders 1 and 4, and the main result has a Lean formalization. The manuscript also shows the complete Barker-sequence length classification for lengths greater than one, using the matrix result together with classical results.

Limits and assumptions

The matrix statement concerns real circulant Hadamard matrices, not arbitrary matrices or Hadamard matrices without the circulant condition. The sequence classification concerns binary sequences with every nontrivial aperiodic autocorrelation bounded in magnitude by 1, and is stated for lengths greater than one. The main matrix result has a Lean formalization; the supplied material does not specify whether that formalization also covers the full Barker-sequence consequence. No proof strategy, detailed formalization scope, companion manuscripts, or remaining open questions are supplied.

Mini glossary

Hadamard matrix: A square matrix of plus and minus ones in which any two distinct rows have inner product zero.
Circulant: Having successive rows obtained by circularly shifting one row, with entries wrapping around.
Order: The number of rows, equivalently columns, of a square matrix.
Aperiodic autocorrelation: The sum of products of corresponding entries when a sequence is compared with a shifted copy, counting only their overlap.
Barker sequence: A sequence of plus and minus ones whose aperiodic autocorrelations at all nonzero overlapping shifts have magnitude at most 1.
Lean formalization: A mathematical statement and its proof expressed for checking by the Lean proof assistant.

Keywords : Circulant Hadamard matrices · Barker sequences · Lean formalization

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves that real circulant Hadamard matrices exist exactly in orders 1 and 4, resolving the circulant Hadamard conjecture. Together with classical Barker-sequence results, this shows that binary sequences whose nontrivial aperiodic autocorrelations have magnitude at most 1 exist at lengths n > 1 exactly when $n \in {2, 3, 4, 5, 7, 11, 13}$ .

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

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lean/docs/179.md

Manuscripts in this family

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The circulant Hadamard conjecture

23/09/2026

Verified by Lean

We prove the circulant Hadamard conjecture: a real circulant Hadamard matrix has order 1 or 4. As a consequence, Barker sequences of length greater than one exist exactly at lengths 2, 3, 4, 5, 7, 11, 13, proving the Barker-sequence conjecture.

PDF GitHub folder Lean formalization