A Hamada type characterization of the classical geometric designs

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The dimension of a combinatorial design D over a finite field F = GF(q) was defined in (Tonchev, Des Codes Cryptogr 17:121–128, 1999) as the minimum dimension of a linear code over F that contains the blocks of D as supports of nonzero codewords. There it was proved that, for any prime power q and any integer n ≥ 2, the dimension over F of a design D that has the same parameters as the complement of a classical point-hyperplane design PGn-1(n, q) or AGn-1(n, q) is greater than or equal to n + 1, with equality if and only if D is isomorphic to the complement of the classical design. It is the aim of the present paper to generalize this Hamada type characterization of the classical point-hyperplane designs in terms of associated codes over F = GF(q) to a characterization of all classical geometric designs PGd (n, q), where 1 ≤ d ≤ n − 1, in terms of associated codes defined over some extension field E = GF(qt ) of F. In the affine case, we conjecture an analogous result and reduce this to a purely geometric conjecture concerning the embedding of simple designs with the parameters of AGd (n, q) into PG(n, q). We settle this problem in the affirmative and thus obtain a Hamada type characterization of AGd (n, q) for d = 1 and for d > (n − 2)/2.

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© Springer Science+Business Media, LLC 2011. Publisher’s version of record: https://doi.org/10.1007/s10623-011-9580-3

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Designs, Codes and Cryptography