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New Hadamard Matrices Found A team led by Levent Alpöge, with the help of Claude, has constructed 12 previously unknown Hadamard matrices of size +1 and -1. On August 12, Alpöge published a string of 23,828 "+" and "-" characters and a decoder for it. Together, they provide twelve Hadamard matrices for all remaining unknown valid orders up to 2000, including order 668. The Hadamard matrix is a square table of plus and minus signs where any two different rows, when paired and added, give zero. Its order is the number of rows and columns. The Hadamard hypothesis asserts that such a matrix exists for every size that is a multiple of four. The order 668 was the smallest case for which its existence remained unknown: the previous case, order 428, was closed by mathematicians in 2004. The FrontierMath benchmark had set the task of building a 668x668 table. In the spring FrontierMath problem on hypergraphs, the problem author confirmed the solution found in work with GPT-5.4 Pro, and began preparing an article. The new result's verification chain starts with a ready-made object: Alpöge's string and the response with the decoder program. 🔗 Read original →