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A Kakeya set in the linear representationT  2  *    (  C  )    ,   C   a nonsingular conic, is the point set covered by a set of    q  +  1    lines, one through each point of   C  . In this article, we classify the small Kakeya sets inT  2  *    (  C  )    . The smallest Kakeya sets have size3   q  2   +  2  q   4    , and all Kakeya sets with weight less than3  (   q  2   −  1  )   4    +  q    are classified: there are approximatelyq  2types.

The small Kakeya sets in T 2 * ( C ) , C a conic