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Journal ArticleDOI

Association schemes, orthogonal arrays and codes from non-degenerate quadrics and hermitian varieties in finite projective geometries

01 Jan 1990-Calcutta Statistical Association Bulletin (North Carolina State University. Dept. of Statistics)-Vol. 40, pp 89-96

Abstract: ABSTRACT: In this paper, coexistence of and relations between association schemes, orthogonal arrays and certain families of projective codes have been examined. The projective codes considered are linear spans of a nice projective set P in a hyperplane H = PG (N -1, s)-such as a quadric or a quadric with its nucleus of polarity or a Hermitian variety. There are two ways to construct association schemes from a projective code. One due to Delsarte (1973) considers the restriction of the Hamming scheme to the code with m weights and if it satisfies Delsarte's condition, an m-class association scheme is obtained by defining two codewords to be i-th associates if the Hamming distance between them is i, i = 0, 1, …, m. The alternative approach, first used by Ray-Chaudburi (1959) and later generalized by Mesner (1967) is to classify points (according to some geometrical criterion) in H = PG(N-1, s) with reference to P. into m types (say). Then, two points of the affine space EG(N, s) (for which H is the hyperplane at infinity) are defined to be i-th associates if the line joining the two points meet H at a point of type i, i = 1, …, m. In many cases, the two association schemes defined with respect to the same projective set have the same parameters. But Examples are given where they do not coincide and, in fact, there are cases where one scheme exists but the other does not.
Topics: Hermitian matrix (53%), Association scheme (53%)
References
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Journal ArticleDOI
R. Calderbank1, William M. Kantor2Institutions (2)
Abstract: On etudie les relations entre les codes [n,k] lineaires a deux poids, les ensembles (n,k,h 1 h 2 ) projectifs et certains graphes fortement reguliers

525 citations


"Association schemes, orthogonal arr..." refers background in this paper

  • ...The relations of these codes to orthogonal arrays and difference sets are described in Calderbank and Kantor (1986), and Chakravarti (1990)....

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Journal ArticleDOI
TL;DR: A method is suggested to calculate the weight distributions of cosets of a code and a “dual concept” of that of perfect codes is presented and examined in detail.
Abstract: Starting from the distance distribution of an unrestricted code and its Mac Williams transform, one defines four parameters that, in the linear case, reduce to the minimum weight and the number of distinct weights of the given code and of its dual. In the general case, one exhibits the combinatorial meaning of these parameters and, using them, one obtains various results on the distance properties of the code. In particular, a method is suggested to calculate the weight distributions of cosets of a code. A “dual concept” of that of perfect codes is also presented and examined in detail.

285 citations


"Association schemes, orthogonal arr..." refers background in this paper

  • ...An answer to the question "when is the restriction of the Hamming Association schemes to a code C itself an association scheme ?" has been provided by Delsarte (1973) for linear codes....

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Journal ArticleDOI
Abstract: Orthogonal arrays can be regarded as natural generalizations of orthogonal Latin squares, and are useful in various problems of experimental design. In this paper the known upper bounds for the maximum possible number of constraints for arrays of strength 2 and 3 have been improved, and certain methods for constructing these arrays have been given.

278 citations


"Association schemes, orthogonal arr..." refers background in this paper

  • ...No three columns of Mare linearly dependent and hence taking all linear combinations of the coordinate vectors w~ get an orthogonal array OA(s 8 , s + 2, s, 3) of strength 3 and index unity (Bose and Bush, 1952). This is also a maximum distance separable code C with n=s+2, k=3, d=n- k + 1 = s (see for instance, MacWilliams and Sloane (1977))....

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  • ...No three columns of Mare linearly dependent and hence taking all linear combinations of the coordinate vectors w~ get an orthogonal array OA(s 8 , s + 2, s, 3) of strength 3 and index unity (Bose and Bush, 1952)....

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Abstract: in whole or in part is permitted for any purpose of the United states Government.

250 citations


Journal ArticleDOI
TL;DR: C over GF(4) which have even weights and have the same weight distribution as the dual code C⊥ are studied, and an upper bound obtained on the minimum distance is obtained.
Abstract: This paper studies codes C over GF(4) which have even weights and have the same weight distribution as the dual code C⊥. Some of the results are as follows. All such codes satisfy C ⊥ = C (If C⊥= C, T has a binary basis.) The number of such C's is determined, and those of length ⩽14 are completely classified. The weight enumerator of C is characterized and an upper bound obtained on the minimum distance. Necessary and sufficient conditions are given for C to be extended cyclic. Two new 5-designs are constructed. A generator matrix for C can be taken to have the form [I | B], where B ⊥ = B . We enumerate and classify all circulant matrices B with this property. A number of open problems are listed.

161 citations