Abstract:The binary Type~I self-dual [26,13,6] code is classical. We give what appears to be the first direct non-computational proof of its uniqueness. We first determine the weight enumerators of the code and its shadow. Degree-one harmonic MacWilliams identities supply the required 1-designs. Elementary intersection counts then show that the two minimal half-shadows each contain 13 words; these words label the 26 coordinates as 13 points and 13 lines, and the two shadow classes become the point-stars and line-stars. From this structure we give two uniqueness proofs: one reconstructs the projective plane of order 3 and the plane code, including the full automorphism group PGL(3,3):2; the other deletes an intrinsic flag, obtains the odd Golay code together with a deep hole coset, and reconstructs the length 26 code and the size of the automorphism group from this coset datum. Thus the natural length-24 object behind the code is the odd Golay code together with its unique orbit of deep hole cosets.
From: Gerald Höhn [view email]
[v1]
Tue, 23 Jun 2026 13:38:29 UTC (14 KB)
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