# A new lower bound for the smallest complete (k, n)-arc in $$\mathrm {PG}(2,q)$$

• S. Alabdullah
• J. W. P. Hirschfeld
Open Access
Article
Part of the following topical collections:

## Abstract

In $$\mathrm {PG}(2,q)$$, the projective plane over the field $$\mathbf{F}_{q}$$ of q elements, a (kn)-arc is a set $$\mathcal {K}$$ of k points with at most n points on any line of the plane. A fundamental question is to determine the values of k for which $$\mathcal {K}$$ is complete, that is, not contained in a $$(k+1,n)$$-arc. In particular, what are the smallest and largest values of k for a complete $$\mathcal {K}$$, denoted by $$t_n(2,q)$$ and $$m_n(2,q)$$? Here, a new lower bound for $$t_n(2,q)$$ is established and compared to known values for small q.

## Keywords

Finite projective plane Arc Lower bound

51E20

## 1 Introduction and background

A projective plane of order q consists of a set of $$q^2+q+1$$ points and a set of $$q^2+q+1$$ lines, where each line contains exactly $$q+1$$ points and two distinct points lie on exactly one line. It follows from the definition that each point is contained in exactly $$q+1$$ lines and two distinct lines have exactly one common point.

The main focus of this paper is to find a lower bound for k of a (kn)-arc in $$\mathrm {PG}(2,q)$$. First, some basic constants and their properties are summarised. See [8, Chap. 12] or [7, Chap. 12].

### Definition 1.1

A (kn)-arc in $$\mathrm {PG}(2,q)$$ is a set $$\mathcal {K}$$ of k points, no $$n+1$$ of which are collinear, but with at least one set of n points collinear. When $$n =2$$, a (k, 2)-arc is a k-arc.

### Definition 1.2

A (kn)-arc is complete if it is not contained in a $$(k,n+1)$$-arc.

### Notation 1.3

The maximum value of k for a (kn)-arc to exist is denoted by $$m_n(2,q)$$.

### Definition 1.4

A line $$\ell$$ is an i-secant of $${\mathcal {K}}$$ if $$|\ell \cap {\mathcal {K}}| =i$$.

### Notation 1.5

For a (kn)-arc $$\mathcal {K}$$ in $$\mathrm {PG}(2,q),$$ let
\begin{aligned} \tau _i= & {} \hbox {the total number of { i}-secants of} \mathcal {K},\\ \rho _i=\rho _i(P)= & {} \hbox {the number of { i}-secants through a point { P} of} \mathcal {K},\\ \sigma _i=\sigma _i(Q)= & {} \hbox {the number of} { i}-\hbox {secants through a point} { Q} \hbox {of } \mathrm {PG}(2,q)\backslash \mathcal {K}. \end{aligned}

### Lemma 1.6

For a (kn)-arc $$\mathcal {K},$$ the following equations hold :
\begin{aligned} \sum _{i=0}^{n} \tau _i= & {} q^2+q+1; \end{aligned}
(1.1)
\begin{aligned} \sum _{i=1}^{n} i\tau _i= & {} k(q+1); \end{aligned}
(1.2)
\begin{aligned} \sum _{i=2}^{n} {\textstyle \frac{1}{2}}i(i-1)\tau _i= & {} {\textstyle \frac{1}{2}}k(k-1). \end{aligned}
(1.3)

### Proof

See [8, Chap. 12]. $$\square$$

The constants $$\rho _i,\,\sigma _i$$ are useful in investigations of the properties of (kn)-arcs, but are not required here.

### Theorem 1.7

\begin{aligned} m_2(2,q)= {\left\{ \begin{array}{ll} q+2, &{} {for}\, q\, {even};\\ q+1, &{} {for}\, q\, {odd}. \end{array}\right. } \end{aligned}

### Proof

See [8, Chap. 8]. $$\square$$

### Theorem 1.8

1. (1)
\begin{aligned} m_n(2,q) {\left\{ \begin{array}{ll} =(n-1)q+n, &{} {for}\, q\, {even}\, {and}\, n\mid \, q;\\ <(n-1)q+n, &{} {for}\, q \,{odd}. \end{array}\right. } \end{aligned}

2. (2)

A (kn)-arc $$\mathcal {K}$$ is maximal if and only if every line in $$\mathrm {PG}(2,q)$$ is either an n-secant or a 0-secant.

### Proof

See [8, Chap. 12]. $$\square$$

### Lemma 1.9

If $$\mathcal {K}$$ is a complete (kn)-arc,  then $$(q+1 - n)\tau _n \ge q^2+q+1-k,$$ with equality if and only if $$\sigma _n=1$$ for all Q in $$\mathrm {PG}(2,q)\backslash \mathcal {K}$$.

### Proof

See [8, Chap. 12]. $$\square$$

### Definition 1.10

The type of a point P in $$\mathrm {PG}(2,q)$$ for a (kn)-arc is the $$(n+1)$$-tuple $$(\rho _0,\rho _1, \ldots ,\rho _n)$$.

## 2 New lower bound

A lower bound for the smallest complete (kn)-arcs $$\mathcal {K}$$ is established below.

### Theorem 2.1

In $$\mathrm {PG}(2,q),$$ a complete (kn)-arc does not exist for $$k \le n^*,$$ where
\begin{aligned} n^*= \frac{(q+1-n^2)+\sqrt{(q+1-n^2)^2+4(n^2-n)(q+1-n)(q^2+q+1)}}{2(q+1-n)}. \end{aligned}

### Proof

Let $$\mathcal {K}$$ be a complete (kn)-arc. The number of n-secants through a point P in $$\mathcal {K}$$ is at most $$(k-1)/(n-1)$$. Then, counting the set $$\{(P,\ell ) \}$$, where $$\ell$$ is an n-secant and P is a point of $$\mathcal {K}$$ incident with $$\ell$$ gives that
\begin{aligned} \tau _n \le \frac{k(k-1)}{n(n-1)}. \end{aligned}
(2.1)
On the other hand, Lemma 1.9 implies that
\begin{aligned} \tau _n \ge \frac{q^2+q+1-k}{q+1-n} \end{aligned}
(2.2)
Now, from Eqs. (2.1) and (2.2),
\begin{aligned} \frac{k^2-k}{n^2-n} = \frac{q^2+q+1-k}{q+1-n}. \end{aligned}
Hence
\begin{aligned}&(q+1-n)k^2-(q+1-n)k = (n^2-n)(q^2+q+1)-(n^2-n)k, \nonumber \\&(q+1-n)k^2-(q+1-n-n^2+n)k -(n^2-n)(q^2+q+1)= 0, \nonumber \\&(q+1-n)k^2-(q+1-n^2)k -(n^2-n)(q^2+q+1)= 0. \end{aligned}
(2.3)
Now, Eq. (2.3) implies that $$k= n^* > 0.$$$$\square$$
This can be applied to k-arcs and (k, 3)-arcs, as in Tables  1 and 2, with the notation $$n^* = b_n(2,q)$$ and $$n=2,3$$.
Table 1

Bounds for complete k-arcs for $$4\le q\le 23$$

q

4

5

7

8

9

11

13

16

17

19

23

$$b_2(2,q)$$

5

5

5

5

6

6

6

7

7

7

8

$$t_2(2,q)$$

6

6

6

6

6

7

8

9

10

10

10

$$m_2(2,q)$$

6

6

8

10

10

12

14

18

20

20

25

Table 2

Bounds for complete (k, 3)-arcs for $$4\le q\le 16$$

q

4

5

7

8

9

11

13

16

$$b_3(2,q)$$

7

8

9

9

9

10

11

12

$$t_3(2,q)$$

7

9

9

11

12

13

15

15

$$m_3(2,q)$$

9

11

15

15

17

21

23

28

## 3 Comparison with known results

Table 3 gives the comparison, for (k, 3)-arcs, between [9, 14] and Theorem 2.1 for the values of q with $$4\le q \le 16$$.
Table 3

Lower bounds for complete (k, 3)-arcs for $$4\le q\le 16$$

k

[14]

[9]

Theorem

Exact result

4

7

6

7

7

5

8

6

8

9

7

9

7

9

9

8

9

8

9

11

9

10

8

10

12

11

10

9

10

13

13

11

10

11

15

16

12

11

12

15

## Notes

### Acknowledgements

Salam Alabdullah obtained a Ph.D. studentship funded by the Ministry of Higher Education and Scientific Research of the Government of Iraq via the University of Basra.

## References

1. 1.
Bartoli D., Marcugini S., Pambianco F.: The maximum and the minimum sizes of complete $$(n,3)$$-arcs in $$PG(2,16)$$. In: Thirteenth International Workshop on Algebraic and Combinatorial Coding Theory, Pomorie, Bulgaria, pp. 77–82, 15–21 June 2012.Google Scholar
2. 2.
Bartoli D., Giulietti M., Zini G.: Complete $$(k,3)$$-arcs from quartic curves. Des. Codes Cryptogr. 79, 487–505 (2015).
3. 3.
Bierbrauer J.: The maximal size of a 3-arc in $$\text{ PG }(2,8)$$. J. Comb. Math. Comb. Comput. 45, 145–161 (2003).
4. 4.
Coolsaet K., Sticker H.: A full classification of the complete $$k$$-arcs in $$PG(2,23)$$ and $$PG(2,25)$$. J. Comb. Des. 17, 459–477 (2009).
5. 5.
Coolsaet K., Sticker H.: The complete $$k$$-arcs of $${\rm PG}(2,27)$$ and $${\rm PG}(2,29)$$. J. Comb. Des. 19, 111–130 (2011).
6. 6.
Coolsaet K., Sticker H.: The complete $$(k,3)$$-arcs of $$\text{ PG }(2, q), q\le 13$$. J. Comb. Des. 20, 89–111 (2012).
7. 7.
Hirschfeld J.W.P.: Projective Geometries Over Finite Fields, pp. xii+474. Clarendon Press, Oxford (1979).Google Scholar
8. 8.
Hirschfeld J.W.P.: Projective Geometries Over Finite Fields, 2nd edn, pp. xiv+555. Oxford University Press, Oxford (1998).Google Scholar
9. 9.
Hirschfeld J.W.P., Pichanick E.V.D.: Bounds for arcs of arbitrary degree in finite Desargusian planes. J. Comb. Des. 24, 184–196 (2016).
10. 10.
Marcugini S., Milani A., Pambianco F.: A computer based classification of the $$(n,3)$$-arcs in $$PG(2,7)$$. Rapporto Tecnico n.7 (1998).Google Scholar
11. 11.
Marcugini S., Milani A., Pambianco F.: A computer based classification of the $$(n,3)$$-arcs in $$PG(2,8)$$. Rapporto Tecnico n.8 (1998).Google Scholar
12. 12.
Marcugini S., Milani A., Pambianco F.: A computer based classification of the $$(n,3)$$-arcs in $$PG(2,9)$$. Rapporto Tecnico n.9 (1998).Google Scholar
13. 13.
Marcugini S., Milani A., Pambianco F.: Maximal $$(n,3)$$-arcs in $$PG(2,11)$$. Discret. Math. 208(209), 421–426 (1999).
14. 14.
Marcugini S., Milani A., Pambianco F.: Maximal $$(n,3)$$-arcs in $$PG(2,13)$$. Discret. Math. 294, 139–145 (2005).
15. 15.
Marcugini S., Milani A., Pambianco F.: Complete arcs in $$PG(2,25)$$: the spectrum of size and the classification of the smallest complete arcs. Discret. Math. 307, 739–747 (2007).