By Dijen Ray-Chaudhuri, Akos Seress

This quantity comprises 4 surveys and 17 refereed examine articles on coding concept, layout idea, organization schemes, algebraic graph idea, combinational geometry and community flaws. it truly is aimed toward researchers and graduate scholars drawn to advancements in coding and layout conception.

**Read or Download Codes and Designs: Proceedings of a Conference Honoring Professor Dijen K. Ray-Chaudhuri on the Occasion of His 65th Birthday, the Ohio State University, May 18-21 PDF**

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Proof. Suppose that (20) holds for all j . Multiply (20) by 2j −k and add the resulting equations for j = k, . . , j = k + d − 1, we obtain that νk (s) = 2d cd+k−1 − ck−1 = (2d − 1)ck−1 . (22) On the p-ranks of GMW difference sets 29 Since ck−1 is an integer, we have that νk (s) ≡ 0 mod 2d −1. This is true for all k since c is an integer sequence, in particular, we have that ν0 (s) = ν(s) ≡ 0 mod 2d − 1. Conversely, it is easily veriﬁed that (20) holds if the cj satisfy (21). The fact that w(c) = w(s) is a trivial consequence of (20); the bounds on the cj are easily veriﬁed.

Cd Un }, with Un being the group of units and ci Un := {ci u : u ∈ Un }. The S-ring corresponding to this association scheme is maximal among rational circulant S-rings [5]. Another important case arises when we take the set of conjugacy classes C = {C0 = {1}, C1 , . . , Cd } as a standard partition for G. The scheme X(C, G) is commutative and is called the group scheme. This association scheme is also a group-case scheme. Indeed, let I (G) be the inner automorphism group of G. Then the group scheme X(C, G) X(G I (G), G) where G I (G) acts on G with G by multiplication and the inner automorphisms in a natural fashion.

3] B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi sums, Wiley Interscience, 1998. [4] T. Beth, D. Jungnickel, and H. 1, Second edition, Cambridge University Press, Cambridge 1999. [5] W. G. Bridges, M. Hall, and J. L. Hayden, Codes and designs, J. Combin. Theory Ser. A 31 (1981), 155–174. [6] H. -S. No, Linear span of extended sequences and cascaded GMW sequences, IEEE Trans. Inform. 6, 2060–2065. [7] R. Evans, H. D. L. Hollmann, C. Krattenthaler, and Q. Xiang, Gauss sums, Jacobi sums and p-ranks of difference sets, J.