Natural bounded concentrators

Abstract

We give the first known direct construction for linear families of bounded concentrators. The construction is explicit and the results are simple natural bounded concentrators.

Let\(\mathbb{F}_q \) be the field withq elements,g(x)F q [x] of degree greater than or equal to 2,\(H = PGL_2 (\mathbb{F}_q )[x]/g(x)\mathbb{F}_q [x]),{\text{ }}B = PGL_2 (\mathbb{F}_q )\) and\(A = \left\{ {\left. {\left( {\begin{array}{*{20}c} a & {b + cx} \\ 0 & 1 \\ \end{array} } \right)} \right|a \in \mathbb{F}_q^* ;b,c \in \mathbb{F}_q } \right\}\). LetI nputs=H/A,O utputs=H/B, and draw an edge betweenaA andbB iffaA∩bB≠ϕ. We prove that for everyq≥5 this graph is an\(\left( {\left| {H/A} \right|,\frac{q}{{q + 1}},q + 1,\frac{{q - 4}}{{q - 3}}} \right)\) concentrator.

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Part of this research was done while the author was at the department of Computer Science, The University of British Columbia, Vancouver, B.C., Canada.

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Morgenstern, M. Natural bounded concentrators. Combinatorica 15, 111–122 (1995). https://doi.org/10.1007/BF01294463

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Mathematics Subject Classification (1991)

  • Primary: 05 C 35
  • Secondary: 05 C 25