Maximal Arc - Definition

Definition

Let be a finite projective plane of order q (not necessarily desarguesian). Maximal arcs of degree d ( 2 ≤ dq- 1) are (k,d)-arcs in, 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 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.

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