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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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