The Paley Design at 343: Self-Dual Codes and Non-Abelian Frobenius-Kernel Obstructions

This record contains the public preprint for the manuscript: “The Paley Design at 343: Self-Dual Codes and Non-Abelian Frobenius-Kernel Obstructions” The paper studies the Paley design PD(343), its associated binary code, its one-coordinate Type II self-dual extension, and a non-abelian Frobenius-kernel obstruction involving the extraspecial group 7^{1+2}. The main results include generator independence of the quadratic-residue construction over GF(7^3), the code parameters C_2(PD(343)) = [343,172,d]_2 with d <= 88, the construction of a self-dual Type II extended code [344,172,d']_2, a computationally found codeword of weight 48 giving d' <= 48, and a proof that Key--Moori Method 1 cannot produce a 2-(343,171,85) design from the extraspecial kernel 7^{1+2} because no complement of order 171 exists in its automorphism group. This is a public preprint/reproducibility release. The exact minimum distance of the extended code remains open unless certified in a later version.