Maximal arc

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. 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. Let π {displaystyle pi } be a finite projective plane of order q (not necessarily desarguesian). Maximal arcs of degree d ( 2 ≤ d ≤ q- 1) are (k,d)-arcs in π {displaystyle pi } , 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 {displaystyle pi } 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 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 ≤ d ≤ q- 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. One can construct partial geometries, derived from maximal arcs: