A Hamada type characterization of the classical geometric designs

作者：Jungnickel Dieter^*; Tonchev Vladimir D

来源：Designs, Codes and Cryptography, 2012, 65(1-2): 15-28.

DOI：10.1007/s10623-011-9580-3

• Classical design
• Hamada conjecture
• Trace code

The dimension of a combinatorial design 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 as supports of nonzero codewords. There it was proved that, for any prime power q and any integer n a parts per thousand yen 2, the dimension over F of a design that has the same parameters as the complement of a classical point-hyperplane design PG (n-1)(n, q) or AG (n-1)(n, q) is greater than or equal to n + 1, with equality if and only if 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 PG (d) (n, q), where 1 a parts per thousand currency sign d a parts per thousand currency sign n - 1, in terms of associated codes defined over some extension field E = GF(q (t) ) 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 AG (d) (n, q) into PG(n, q). We settle this problem in the affirmative and thus obtain a Hamada type characterization of AG (d) (n, q) for d = 1 and for d %26gt; (n - 2)/2.

• 出版日期2012-10