On the maximum size of some \((k,r)\)-arcs in PG\((2,q)\). (English) Zbl 1142.51004
A \((k,r)\)-arcin PG\((2,q)\) is a set of \(k\) points, such that some \(r\), but no \(r+1\) of them, are collinear. The maximum size of a \((k,r)\)-arc in PG\((2,q)\) is denoted by \(m_r(2,q)\). S. Ball [J. Lond. Math. Soc. (2) 54, No. 3, 581–593 (1996; Zbl 0904.51002)] proved that in PG\((2,q)\), \(q\) prime, \(m_r(2,q)\leq (r-1)q+r-(q+1)/2\) for \(r\geq (q+3)/2\). For \(r=(q+3)/2\), this bound is sharp.
In this article, it is proven that \(m_r(2,q)\leq (r-1)q+r-(q+3)/2\), for \(r>(q+3)/2\) and \(q=17,19,23,29\). As an application of these results, the results are interpreted for linear codes in relation with the Griesmer lower bound. The Griesmer bound states a lower bound \(g_q(k,d)\) on the length of a linear \([n,k,d]\)-code over the finite field \(\mathbb{F}_q\) of order \(q\).
The results presented here imply that there do not exist linear codes (meeting the Griesmer bound) with parameters \([(r-1)q+r-(q+1)/2,3,(r-1)q-(q+1)/2]_q\), for \((q+3)/2<r<q\) and \(q=7,11,13,17,19,23,29\).
In this article, it is proven that \(m_r(2,q)\leq (r-1)q+r-(q+3)/2\), for \(r>(q+3)/2\) and \(q=17,19,23,29\). As an application of these results, the results are interpreted for linear codes in relation with the Griesmer lower bound. The Griesmer bound states a lower bound \(g_q(k,d)\) on the length of a linear \([n,k,d]\)-code over the finite field \(\mathbb{F}_q\) of order \(q\).
The results presented here imply that there do not exist linear codes (meeting the Griesmer bound) with parameters \([(r-1)q+r-(q+1)/2,3,(r-1)q-(q+1)/2]_q\), for \((q+3)/2<r<q\) and \(q=7,11,13,17,19,23,29\).
Reviewer: Leo Storme (Gent)
MSC:
| 51E20 | Combinatorial structures in finite projective spaces |
| 05B25 | Combinatorial aspects of finite geometries |
| 94B05 | Linear codes (general theory) |
Citations:
Zbl 0904.51002References:
| [1] | Ball, S., On sets of points in finite planes (1994), Ph.D. Thesis: Ph.D. Thesis University of Sussex |
| [2] | Ball, S., Multiple blocking sets and arcs in finite planes, J. London Math. Soc., 54, 427-435 (1996) |
| [3] | Barlotti, A., Some topics in finite geometrical structures, Mimeo series, 439 (1965), Institute of Statistics: Institute of Statistics University of Carolina |
| [4] | Hill, R., Optimal linear codes, (Mitchell, C., Cryptography and Coding (1992), Oxford University Press: Oxford University Press Oxford), 75-104 · Zbl 0742.94012 |
| [5] | Tallini Scafati, M., Sui \((k, n)\)-archi di un piano grafico finito, Rend. Naz. Lincei (8), 49, 1-6 (1966) · Zbl 0146.41702 |
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