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The difference matrices of the classes of a Sharma-Kaushik partition. (English) Zbl 1114.05101
Sharma-Kaushik partition (SK-partition) introduced by B. D. Sharma and M. L. Kaushik [41st Annual Conf. Intern. Stat. Inst., New Delhi 1977] is a special partition of the set of all residue classes \(F_q\) modulo \(q\) (\(q\) is a positive integer \(\geq 2\)); \(F_q=\{0,1,\dots ,q-1\}\). For an SK-partition \(P=\{B_0, B_1, \dots ,B_{m-1}\}\) of \(F_q\) the authors define the difference matrices \(\overline B_i\) (\(0\leq i\leq m-1\)) of \(B_i\) as follows \[ (x,y)\text{\;entry of\;} \overline B_i=\begin{cases} 1& \text{if}\quad x-y\in B_i \\ 0& \text{otherwise.} \end{cases} \]
The authors show that these difference matrices are circulant matrices and derive a series of theorems on them.
The final section of this paper is devoted to the Bose-Mesner algebra of an SK-partition \(P=\{B_0, B_1, \dots ,B_{m-1}\}\) of \(F_q\), which is the algebra generated by the matrices \(\overline B_i\).
MSC:
05E30 Association schemes, strongly regular graphs
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
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