GPT-6 Astra raises MOLS lower bounds at 33 orders, and order 22 finally reaches four squares

Lower bounds on MOLS rise at 33 orders between 22 and 494, and order 22 yields 4 mutually orthogonal Latin squares for the first time. In the 2024 compilation, these orders had lower bounds such as 3 at order 22, 7 at order 427, and 6 each at orders 430 and 494. The table compares old and new bounds order by order, with gains from 1 to 5; the largest gain, 5, appears at orders 427, 430 and 494.

The constructions follow existing routes in combinatorial design rather than searching whole Latin squares from scratch. These routes had already been used for small-order MOLS, including difference matrices, quasi-difference matrices, lifting coarse orthogonal arrays and selecting points from a projective plane. Each order in the table is labeled with its construction, for example difference matrices at orders 45, 72, 80 and 96, lifted coarse orthogonal arrays at orders 50, 54 and 98, and projective planes at several orders from 175 to 494.

The 4 MOLS of order 22 also supply a strongly regular graph whose existence had been open. A strongly regular graph with parameters (484,126,40,30) had been listed as an open case. Under the standard construction, the 484 vertices correspond to cells of a Latin square, with two cells adjacent when they share a row, a column, or a symbol in one of the 4 squares; the adjacency matrix is provided with the construction.

The results cover 33 orders between 22 and 494, with each MOLS set and its verification code in a public repository; the 4 MOLS of order 22, combined with existing results, give at least 4 MOLS for every order from n=22 upward.

These are lower-bound improvements, and upper bounds do not tighten with them; the number of MOLS at order 10 remains undetermined, and even the search space for 3 MOLS is too large to search completely.