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| A '''Maximal arc''' in a finite [[projective plane]] is a largest possible (''k'',''d'')-[[Arc (projective geometry)|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.
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| ==Definition==
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| Let <math>\pi</math> be a finite projective plane of order ''q'' (not necessarily [[Desargues' theorem|desarguesian]]). Maximal arcs of ''degree'' ''d'' ( 2 ≤ ''d'' ≤ ''q''- 1) are (''k'',''d'')-[[Arc (projective geometry)|arcs]] in <math>\pi</math>, where ''k'' is maximal with respect to the parameter ''d'', in other words, ''k'' = ''qd'' + ''d'' - ''q''.
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| Equivalently, one can define maximal arcs of degree ''d'' in <math>\pi</math> as non-empty sets of points ''K'' such that every line intersects the set either in 0 or ''d'' points.
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| Some authors permit the degree of a maximal arc to be 1, ''q'' or even ''q''+ 1.<ref>{{harvnb|Hirschfeld|1979|loc=pp. 325}}</ref> Letting ''K'' be a maximal (''k'', ''d'')-arc in a projective plane of order ''q'', if
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| * ''d'' = 1, ''K'' is a point of the plane,
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| * ''d'' = ''q'', ''K'' is the complement of a line (an [[projective plane#Affine planes|affine plane]] of order ''q''), and
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| * ''d'' = ''q'' + 1, ''K'' is the entire projective plane.
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| 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.
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| ==Properties==
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| * The number of lines through a fixed point ''p'', not on a maximal arc ''K'', intersecting ''K'' in ''d'' points, equals <math>(q+1)-\frac{q}{d}</math>. Thus, ''d'' divides ''q''.
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| * In the special case of ''d'' = 2, maximal arcs are known as [[Oval (projective plane)|hyperovals]] which can only exist if ''q'' is even.
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| * 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.<ref>{{harvnb|Hirschfeld|1979|loc=pg. 328}}</ref>
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| * In PG(2,''q'') with ''q'' odd, no non-trivial maximal arcs exist.<ref>{{harvnb|Ball|Blokhuis|Mazzocca|1997}}</ref>
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| * In PG(2,2<sup>''h''</sup>), maximal arcs for every degree 2<sup>''t''</sup>, 1 ≤ ''t'' ≤ ''h'' exist.<ref>{{harvnb|Denniston|1969}}</ref>
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| ==Partial geometries==
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| One can construct [[partial geometry|partial geometries]], derived from maximal arcs:<ref>{{harvnb|Thas|1974}}</ref>
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| * Let ''K'' be a maximal arc with degree ''d''. Consider the incidence structure <math>S(K)=(P,B,I)</math>, 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 : <math>pg(q-d,q-\frac{q}{d},q-\frac{q}{d}-d+1)</math>.
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| * Consider the space <math>PG(3,2^h) (h\geq 1)</math> and let ''K'' a maximal arc of degree <math>d=2^s (1\leq s\leq m)</math> in a two-dimensional subspace <math>\pi</math>. Consider an incidence structure <math>T_2^{*}(K)=(P,B,I)</math> where ''P'' contains all the points not in <math>\pi</math>, ''B'' contains all lines not in <math>\pi</math> and intersecting <math>\pi</math> in a point in ''K'', and ''I'' is again the natural inclusion. <math>T_2^{*}(K)</math> is again a partial geometry : <math>pg(2^h-1,(2^h+1)(2^m-1),2^m-1)\,</math>.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{citation|last1=Ball|first1=S.|last2=Blokhuis|first2=A.|last3=Mazzocca|first3=F.|title=Maximal arcs in Desarguesian planes of odd order do not exist|journal=Combinatorica|volume=17|year=1997|pages=31–41|mr=98h:51014|zbl=0880.51003}}
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| * {{citation|last=Denniston|first=R.H.F.|title=Some maximal arcs in finite projective planes|journal=J. Combin. Theory|volume=6|year=1969|pages=317–319|mr=39#1345|zbl=0167.49106}}
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| * {{citation|last=Hirschfeld|first=J.W.P.|title=Projective Geometries over Finite Fields|year=1979|publisher=Oxford University Press|location=New York|isbn=0-19-853526-0}}
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| * {{citation|last=Mathon|first=R.|title=New maximal arcs in Desarguesian planes|journal=J. Combin. Theory Ser. A|volume=97|year=2002|pages=353–368|mr=2002k:51012|zbl=01763788}}
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| * {{citation|last=Thas|first=J.A.|title=Construction of maximal arcs and partial geometries|journal=Geom. Dedicata|volume= 3|year=1974|pages=61–64|mr=50#1931|zbl=0285.50018}}
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| {{DEFAULTSORT:Maximal Arc}}
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| [[Category:Projective geometry]]
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