The Association Schemes of Coding Theory

Abstract.A type II matrix is a square matrix W with non-zero complex entries such that the entrywise quotient of any two distinct rows of W sums to zero. Hadamard matrices and character tables of abelian groups are easy examples, and other examples called spin models and satisfying an additional condition can be used as basic data to construct invariants of links in 3-space. Our main result is the construction, for every type II matrix W , of a Bose-Mesner algebra N (W ), which is a commutative algebra of matrices containing the identity I , the all-one matrix J , closed under transposition and under Hadamard (i.e., entrywise) product. Moreover, if W is a spin model, it belongs to N (W ). The transposition of matrices W corresponds to a classical notion of duality for the corresponding Bose-Mesner algebras N (W ). Every Bose-Mesner algebra encodes a highly regular combinatorial structure called an association scheme, and we give an explicit construction of this structure. This allows us to compute N (W ) for a number of examples.

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“…Thus, we agree with the terminology of [10], and our association schemes are commutative in the terminology of [5].…”

Section: Association Schemes and Bose-mesner Algebrassupporting

confidence: 70%

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Jaeger¹,

Matsumoto

Nomura

1998

Journal of Algebraic Combinatorics

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“…The characteristic vector of a subset S in Ω is the vector χ S in RΩ with (χ S ) x = 1 if x ∈ S, and (χ S ) x = 0 if x / ∈ S. In [8], the outer distribution and inner distribution of a non-empty subset S of Ω is introduced. The outer distribution of S is the |Ω| × (d + 1)-matrix B, with B x,i = |{x ∈ S|(x, x ) ∈ R i }|.…”

Section: Association Schemesmentioning

confidence: 99%

“…The terminology and techniques that we will use were developed by Delsarte, and will be explained in Section 2. In [8], he introduced powerful algebraic techniques to study subsets in association schemes (see Subsection 2.1 for the definition). He also developed a general theory of regular semilattices in [9], providing a generalizing notion of t-designs in several association schemes and an algebraic characterization of them.…”

Section: Introductionmentioning

confidence: 99%

Antidesigns and regularity of partial spreads in dual polar graphs

Vanhove

2010

J. Combin. Designs

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We give several examples of designs and antidesigns in classical finite polar spaces. These types of subsets of maximal totally isotropic subspaces generalize the dualization of the concepts of m-ovoids and tight sets of points in generalized quadrangles. We also consider regularity of partial spreads and spreads. The techniques that we apply were developed by Delsarte. In some polar spaces of small rank, some of these subsets turn out to be completely regular codes.

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“…This structures allow one to use the usual association scheme methods to study subsets Y ⊂ X k (cf. [DL98]). ‡ Here, we use the definition of a generalized t-designs as in [Del73]: An element x ∈ K n is said to be covered by an element y ∈ K n if each nonzero component x i of x is equal to the corresponding component y i of y.…”

Section: Constant Weight Codes and Generalized T-designsmentioning

confidence: 99%

Self-dual codes over the Kleinian four group

Höhn

2003

Mathematische Annalen

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The Association Schemes of Coding Theory

Cited by 610 publications

References 18 publications

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Antidesigns and regularity of partial spreads in dual polar graphs

Self-dual codes over the Kleinian four group

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