maximal arc

Let $\backslash pi$ 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 $\backslash pi$, 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 $\backslash pi$ 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.{{harvnb|Hirschfeld|1979|loc=pp. 325}} 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 ≤ 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.

• The number of lines through a fixed point p, not on a maximal arc K, intersecting K in d points, equals $(q+1)-\backslash frac\{q\}\{d\}$. 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.{{harvnb|Hirschfeld|1979|loc=pg. 328}}
• In PG(2,q) with q odd, no non-trivial maximal arcs exist.{{harvnb|Ball|Blokhuis|Mazzocca|1997}}
• In PG(2,2^h), maximal arcs for every degree 2^t, 1 ≤ t ≤ h exist.{{harvnb|Denniston|1969}}

One can construct partial geometries, derived from maximal arcs:{{harvnb|Thas|1974}}

• 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 : $pg(q-d,q-\backslash frac\{q\}\{d\},q-\backslash frac\{q\}\{d\}-d+1)$.
• Consider the space $PG(3,2^h)\; (h\backslash geq\; 1)$ and let K a maximal arc of degree $d=2^s\; (1\backslash leq\; s\backslash leq\; m)$ in a two-dimensional subspace $\backslash pi$. Consider an incidence structure $T\_2^\{*\}(K)=(P,B,I)$ where P contains all the points not in $\backslash pi$, B contains all lines not in $\backslash pi$ and intersecting $\backslash pi$ in a point in K, and I is again the natural inclusion. $T\_2^\{*\}(K)$ is again a partial geometry : $pg(2^h-1,(2^h+1)(2^m-1),2^m-1)\backslash ,$.

{{reflist}}

• {{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=1466573|zbl=0880.51003|doi=10.1007/bf01196129}}
• {{citation|last=Denniston|first=R. H. F.|title=Some maximal arcs in finite projective planes|journal=Journal of Combinatorial Theory|volume=6|issue=3|date=April 1969|pages=317–319|mr=239991|zbl=0167.49106|doi=10.1016/s0021-9800(69)80095-5|doi-access=free}}
• {{citation|last=Hirschfeld|first=J. W. P.|title=Projective Geometries over Finite Fields|year=1979|publisher=Oxford University Press|location=New York|isbn=978-0-19-853526-3|url-access=registration|url=https://archive.org/details/projectivegeomet0000hirs}}
• {{citation|last=Mathon|first=Rudolf|title=New maximal arcs in Desarguesian planes|journal=Journal of Combinatorial Theory | series=Series A|volume=97|issue=2|year=2002|pages=353–368|mr=1883870|zbl=1010.51009|doi=10.1006/jcta.2001.3218|doi-access=free}}
• {{citation|last=Thas|first=J. A.|title=Construction of maximal arcs and partial geometries|journal=Geometriae Dedicata|volume= 3|year=1974|pages=61–64|mr=349437|zbl=0285.50018|doi=10.1007/bf00181361}}

{{DEFAULTSORT:Maximal Arc}}

Category:Projective geometry