Association schemes of quadratic forms

We describe fission schemes of most known classical self-dual association schemes, such as the Hamming scheme H(n, q) when q is a prime power. These fission schemes are themselves self-dual, with the exception of certain quadratic forms schemes in even characteristic.1999 Academic Press 1. THE HAMMING SCHEME The Hamming scheme H(n, q) is defined on vertex set X n of words of length n from an alphabet X of size q. Two words are in relation R i if and only if they differ in precisely i positions (we also say the words have distance i). If q is a prime power, then we can put extra structure on the set X, and this allows us to refine one of the relations of the Hamming scheme. For some background in the theory of association schemes, we refer the reader to [1, Chap. 2]. Theorem 1. Let q be a prime power. Then the Hamming scheme H(n, q) on vertex set GF(q) n has a self-dual fission scheme obtained by splitting the distance-n relation into q&1 relations according to the value of >n such that wt(b&a)=n.Proof. Relations R i , i=0, ..., n&1 from the Hamming scheme remain the same, while R n is split into S x , x # GF(q)*, where (a, b) # S x if and only if > n i=1 (b i &a i )=x. To prove that this defines an association scheme we show that the intersection parameters are well-defined; these come in six Article ID jcta.1999.2970, available online at http:ÂÂwww.idealibrary.com on 167

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“…Write n := m+1 Proof. If the characteristic of K is odd, the theorem was proved by Egawa [18,Thm. 2].…”

Section: Final Remarksmentioning

confidence: 92%

Symmetric bilinear forms over finite fields of even characteristic

Schmidt

2010

Journal of Combinatorial Theory, Series A

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Let S m be the set of symmetric bilinear forms on an m-dimensional vector space over GF(q), where q is a power of two. A subset Y of S m is called an (m, d)-set if the difference of every two distinct elements in Y has rank at least d. Such objects are closely related to certain families of codes over Galois rings of characteristic four.An upper bound on the size of (m, d)-sets is derived, and in certain cases, the rank distance distribution of an (m, d)-set is explicitly given. Constructions of (m, d)-sets are provided for all possible values of m and d.

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Association schemes of quadratic forms

Cited by 35 publications

References 9 publications

Distance-Regular Graphs

Distance-Regular Graphs

Fissions of Classical Self-Dual Association Schemes

Symmetric bilinear forms over finite fields of even characteristic

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