Circulant weighing matrices: the open cases

A circulant weighing matrix is a cyclic matrix of $0$ , $+1$ and $-1$ entries whose rows are mutually orthogonal. Which sizes and weights are possible is settled for most small parameters, but thousands of cases remain open in Dan Gordon's database, the smallest at order 96 and weight 36. A candidate is a single row that can be checked in milliseconds.

Problem

A circulant weighing matrix $CW(n, k)$ is an $n \times n$ matrix $W$ with entries in $\{0, \pm 1\}$ , each row a cyclic shift of the previous one, such that

W W^{\top} = k I_n.

Equivalently, its first row $(a_0, \dots, a_{n-1})$ has exactly $k$ nonzero entries and periodic autocorrelation $\sum_{i} a_i a_{i+t \bmod n} = 0$ for every shift $t \ne 0$ . The weight must be a square, $k = s^2$ , with $(s^2+s)/2$ entries equal to $+1$ and $(s^2-s)/2$ equal to $-1$ . The database focuses on proper matrices, whose nonzero positions do not all lie in a coset of a proper subgroup of $\mathbb{Z}/n\mathbb{Z}$ .

Task: decide existence for the parameter pairs marked open, starting with the smallest: $CW(96, 36)$ , $CW(105, 36)$ , $CW(112, 36)$ , $CW(112, 100)$ , $CW(117, 36)$ and $CW(120, 49)$ . (The dataset's keys use $s$ rather than $k$ , so the first of these is stored as CW(96,6).)

Background

Weighing matrices generalize Hadamard matrices (the case $k = n$ ), and circulant ones are equivalent to group-ring equations over the cyclic group, which ties them to difference sets and relative difference sets. Strassler's 1997 thesis tabulated existence for $n \le 200$ and $k \le 100$ ; Tan (2016) updated the table, and Arasu, Gordon and Zhang (2021) settled 12 of the 34 cases still open then. Gordon's La Jolla database extends the table to $n \le 1000$ and records a construction or a nonexistence argument for each settled pair.

What is known

Database (October 2026): 15,536 parameter pairs with $n \le 1000$ , of which about 2,770 are open; the rest are marked as existing, completely classified or impossible.
Strassler's range ( $n \le 200$ , $k \le 100$ ): 28 pairs remain open in the database, from $CW(96, 36)$ to $CW(198, 81)$ .
Constructions: most known matrices come from smaller ones by a product construction or from relative difference sets, with a few sporadic examples; Gordon (2026) extends the classification of cyclic relative difference sets and applies it to circulant weighing matrices.
Circulant Hadamard matrices ( $k = n$ ): Ryser's conjecture that none exists beyond order 4 is claimed proved in an OpenAI preprint (The circulant Hadamard conjecture, 23 September 2026), whose main theorem is formalized in Lean; it is not yet refereed.
Nonexistence: Arasu, Gordon and Zhang (2021) describe several of the methods used to rule out parameter pairs, alongside earlier work collected in Strassler's and Tan's tables.

What would count as progress

A first row for an open pair, given as the positions of its $+1$ and $-1$ entries. Checking the $n - 1$ autocorrelations takes milliseconds.
A nonexistence proof for an open pair, by an algebraic argument or by an exhaustive search whose code can be rerun.
New general constructions that fill many open entries at once.

References

K. T. Arasu, D. M. Gordon and Y. Zhang, "New nonexistence results on circulant weighing matrices", Cryptography and Communications 13 (2021), 775–789. doi:10.1007/s12095-021-00492-0
D. M. Gordon, "Cyclic relative difference sets and circulant weighing matrices", Journal of Algebraic Combinatorics 63 (2026). doi:10.1007/s10801-026-01515-w
M. M. Tan, "Group invariant weighing matrices" (2016). arXiv:1610.01914
D. M. Gordon, "La Jolla Circulant Weighing Matrices Repository" (dataset, 2024). doi:10.5281/zenodo.10775927

Source

Imported from La Jolla Circulant Weighing Matrices Repository, curated by Daniel M. Gordon. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.