Harman Patil (Editor)

Maximal arc

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A Maximal arc in a finite projective plane is a largest possible (k,d)-arc in that projective plane. If the finite projective plane has order q (there are q+1 points on any line), then for a maximal arc, k, the number of points of the arc, is the maximum possible (= qd + d - q) with the property that no d+1 points of the arc lie on the same line.

Contents

Definition

Let π be a finite projective plane of order q (not necessarily desarguesian). Maximal arcs of degree d ( 2 ≤ dq- 1) are (k,d)-arcs in π , where k is maximal with respect to the parameter d, in other words, k = qd + d - q.

Equivalently, one can define maximal arcs of degree d in π as non-empty sets of points K such that every line intersects the set either in 0 or d points.

Some authors permit the degree of a maximal arc to be 1, q or even q+ 1. Letting K be a maximal (k, d)-arc in a projective plane of order q, if

  • d = 1, K is a point of the plane,
  • d = q, K is the complement of a line (an affine plane of order q), and
  • d = q + 1, K is the entire projective plane.
  • All of these cases are considered to be trivial examples of maximal arcs, existing in any type of projective plane for any value of q. When 2 ≤ dq- 1, the maximal arc is called non-trivial, and the definition given above and the properties listed below all refer to non-trivial maximal arcs.

    Properties

  • The number of lines through a fixed point p, not on a maximal arc K, intersecting K in d points, equals ( q + 1 ) q d . Thus, d divides q.
  • In the special case of d = 2, maximal arcs are known as hyperovals which can only exist if q is even.
  • An arc K having one fewer point than a maximal arc can always be uniquely extended to a maximal arc by adding to K the point at which all the lines meeting K in d - 1 points meet.
  • In PG(2,q) with q odd, no non-trivial maximal arcs exist.
  • In PG(2,2h), maximal arcs for every degree 2t, 1 ≤ th exist.
  • Partial geometries

    One can construct partial geometries, derived from maximal arcs:

  • Let K be a maximal arc with degree d. Consider the incidence structure S ( K ) = ( P , B , I ) , where P contains all points of the projective plane not on K, B contains all line of the projective plane intersecting K in d points, and the incidence I is the natural inclusion. This is a partial geometry : p g ( q d , q q d , q q d d + 1 ) .
  • Consider the space P G ( 3 , 2 h ) ( h 1 ) and let K a maximal arc of degree d = 2 s ( 1 s m ) in a two-dimensional subspace π . Consider an incidence structure T 2 ( K ) = ( P , B , I ) where P contains all the points not in π , B contains all lines not in π and intersecting π in a point in K, and I is again the natural inclusion. T 2 ( K ) is again a partial geometry : p g ( 2 h 1 , ( 2 h + 1 ) ( 2 m 1 ) , 2 m 1 ) .
  • References

    Maximal arc Wikipedia


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