Abstract
We point out an interplay between \(F_q\)-Frobenius non-classical plane curves and complete \(\left( {k,d} \right)\)-arcs in \(P^{\text{2}} \left( {F_q } \right)\). A typical example that shows how this works is the one concerning an Hermitian curve. We present some other examples here which give rise to the existence of new complete \(\left( {k,d} \right)\)-arcs with parameters \(k = d\left( {q - d + 2} \right)\) and \(d = \left( {q - 1} \right)/\left( {q\prime - 1} \right),q\prime\) being a power of the characteristic. In addition, for q a square, new complete \(\left( {k,d} \right)\)-arcs with either \(k = q\sqrt q b + 1\) and \(d = \left( {\sqrt q + 1} \right)b\left( {2 \leqslant b \leqslant \sqrt q - 1} \right)\) or \(k = \left( {q - 1} \right)\sqrt q b + \sqrt q + 1\) and \(d = \left( {\sqrt q + 1} \right)b\left( {2 \leqslant b \leqslant \sqrt q - 2} \right)\) are constructed by using certain reducible plane curves.
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Giulietti, M., Pambianco, F., Torres, F. et al. On Complete Arcs Arising from Plane Curves. Designs, Codes and Cryptography 25, 237–246 (2002). https://doi.org/10.1023/A:1014979211916
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DOI: https://doi.org/10.1023/A:1014979211916