By Jagdish Srivastava

ISBN-10: 0720422620

ISBN-13: 9780720422627

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**Example text**

3 (mod 4). It follows that for these values of q (except for q = 6), D cannot exist. The question whether D can exist when q = 6 is still open. Bose's and Shrikhande's proof essentially depends on the characterization 40 R. C. BOSE CH. 4 theorem for the triangular association scheme. They prove that if an incidence structure D with the properties (i), (ii), (iii), (iv) exists, then there are v = i(q+2)(q+l) lines of type II and \q{q— 1) lines of type I satisfying the following conditions: (a) Any two lines of type I intersect in exactly one point.

Roy. Soc. London A 247, 151-165. J. M. Goethals, and J. J. Seidel, 1967, Orthogonal matrices with zero diagonal, Canad. J. Math. 19, 1001-1010. J. M. Goethals, and J. J. Seidel, 1970, Strongly regular graphs derived from combinatorial designs, Canad. J. Math. 22, 597-614. M. Golay, 1949, Notes on digital coding, Proc. E. 37, 637. A. J. Hoffman, and R. R. M. J. Res. Develop. 4, 497-504. A. J. Hoffman, 1963, On the polynomial of a graph, Am. Math. Monthly 70, 30-36. N. Robertson, (private communication).

The com position of (0, 0, 1, 2, 1, 0, 2, 2, 1, 0, 1, 1) is 0 4 1 5 2 3 ). Lemma. Every word of composition 0 9 1 3 lies in some coset of the (12, 6) ternary Golay code which contains two words of composition 0 9 1 3 , two words of composition 0 9 2 3 , and no other words of weight < 3 . SEIDEL CH. 3 This lemma may be proved by investigating the distribution of words of weight 3 in the cosets of the (12, 6) code; we shall not present the details here. Now the construction runs as follows: It is wellknown that the (12, 6) ternary Golay code contains as a subcode the (12, 1) repetition code whose three vectors have compositions 012, l 12 , and 2 12 .

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