# American Institute of Mathematical Sciences

May  2011, 5(2): 209-223. doi: 10.3934/amc.2011.5.209

## The classification of $(42,6)_8$ arcs

 1 Department of Mathematics, Colorado State University, Fort Collins, CO 80523, United States 2 Department of Mathematics and RINS, Gyeongsang National University, Jinju, 660-701, South Korea, South Korea 3 Department of Mathematics and Information Sciences, Osaka Prefecture University, Sakai, Osaka 599-8531

Received  April 2010 Revised  October 2010 Published  May 2011

It is known that $42$ is the largest size of a $6$-arc in the Desarguesian projective plane of order $8$. In this paper, we classify these $(42,6)_8$ arcs. Equivalently, we classify the smallest $3$-fold blocking sets in PG$(2,8)$, which have size $31$.
Citation: Anton Betten, Eun Ju Cheon, Seon Jeong Kim, Tatsuya Maruta. The classification of $(42,6)_8$ arcs. Advances in Mathematics of Communications, 2011, 5 (2) : 209-223. doi: 10.3934/amc.2011.5.209
##### References:
 [1] S. Ball, Multiple blocking sets and arcs in finite planes, J. London Math. Soc. (2), 54 (1996), 581-593.  Google Scholar [2] A. Betten and D. Betten, There is no Drake/Larson linear space on $30$ points, J. Combin. Des., 18 (2010), 48-70.  Google Scholar [3] A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, A new table of constant weight codes, IEEE Trans. Inform. Theory, 36 (1990), 1334-1380. doi: 10.1109/18.59932.  Google Scholar [4] J. W. P. Hirschfeld, "Projective Geometries over Finite Fields,'' 2nd edition, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar [5] S. M. Johnson, A new upper bound for error-correcting codes, IRE Trans., IT-8 (1962), 203-207.  Google Scholar [6] F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes. II,'' North-Holland Publishing Co., Amsterdam, 1977.  Google Scholar [7] J. R. M. Mason, A class of $((p^n-p^m)(p^n-1),p^n-p^m)$-arcs in PG$(2,p^n)$, Geom. Dedicata, 15 (1984), 355-361.  Google Scholar

show all references

##### References:
 [1] S. Ball, Multiple blocking sets and arcs in finite planes, J. London Math. Soc. (2), 54 (1996), 581-593.  Google Scholar [2] A. Betten and D. Betten, There is no Drake/Larson linear space on $30$ points, J. Combin. Des., 18 (2010), 48-70.  Google Scholar [3] A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, A new table of constant weight codes, IEEE Trans. Inform. Theory, 36 (1990), 1334-1380. doi: 10.1109/18.59932.  Google Scholar [4] J. W. P. Hirschfeld, "Projective Geometries over Finite Fields,'' 2nd edition, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar [5] S. M. Johnson, A new upper bound for error-correcting codes, IRE Trans., IT-8 (1962), 203-207.  Google Scholar [6] F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes. II,'' North-Holland Publishing Co., Amsterdam, 1977.  Google Scholar [7] J. R. M. Mason, A class of $((p^n-p^m)(p^n-1),p^n-p^m)$-arcs in PG$(2,p^n)$, Geom. Dedicata, 15 (1984), 355-361.  Google Scholar
 [1] Ivan Landjev. On blocking sets in projective Hjelmslev planes. Advances in Mathematics of Communications, 2007, 1 (1) : 65-81. doi: 10.3934/amc.2007.1.65 [2] Alice B. Tumpach, Stephen C. Preston. Quotient elastic metrics on the manifold of arc-length parameterized plane curves. Journal of Geometric Mechanics, 2017, 9 (2) : 227-256. doi: 10.3934/jgm.2017010 [3] Víctor Jiménez López, Gabriel Soler López. A topological characterization of ω-limit sets for continuous flows on the projective plane. Conference Publications, 2001, 2001 (Special) : 254-258. doi: 10.3934/proc.2001.2001.254 [4] Michael Kiermaier, Matthias Koch, Sascha Kurz. $2$-arcs of maximal size in the affine and the projective Hjelmslev plane over $\mathbb Z$25. Advances in Mathematics of Communications, 2011, 5 (2) : 287-301. doi: 10.3934/amc.2011.5.287 [5] Héctor A. Tabares-Ospina, Mauricio Osorio. Methodology for the characterization of the electrical power demand curve, by means of fractal orbit diagrams on the complex plane of Mandelbrot set. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1895-1905. doi: 10.3934/dcdsb.2020008 [6] D. G. Aronson, N. V. Mantzaris, Hans Othmer. Wave propagation and blocking in inhomogeneous media. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 843-876. doi: 10.3934/dcds.2005.13.843 [7] Kristian Bjerklöv, Russell Johnson. Minimal subsets of projective flows. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 493-516. doi: 10.3934/dcdsb.2008.9.493 [8] Jungkai A. Chen and Meng Chen. On projective threefolds of general type. Electronic Research Announcements, 2007, 14: 69-73. doi: 10.3934/era.2007.14.69 [9] Liliana Trejo-Valencia, Edgardo Ugalde. Projective distance and $g$-measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3565-3579. doi: 10.3934/dcdsb.2015.20.3565 [10] Roland Hildebrand. Barriers on projective convex sets. Conference Publications, 2011, 2011 (Special) : 672-683. doi: 10.3934/proc.2011.2011.672 [11] Behrouz Kheirfam, Morteza Moslemi. On the extension of an arc-search interior-point algorithm for semidefinite optimization. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 261-275. doi: 10.3934/naco.2018015 [12] Térence Bayen, Marc Mazade, Francis Mairet. Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 39-58. doi: 10.3934/dcdsb.2015.20.39 [13] Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A flame propagation model on a network with application to a blocking problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 825-843. doi: 10.3934/dcdss.2018051 [14] J. C. Alvarez Paiva and E. Fernandes. Crofton formulas in projective Finsler spaces. Electronic Research Announcements, 1998, 4: 91-100. [15] Mickaël Crampon. Entropies of strictly convex projective manifolds. Journal of Modern Dynamics, 2009, 3 (4) : 511-547. doi: 10.3934/jmd.2009.3.511 [16] Demetres D. Kouvatsos, Jumma S. Alanazi, Kevin Smith. A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 781-816. doi: 10.3934/naco.2011.1.781 [17] Kazuhisa Ichikawa. Synergistic effect of blocking cancer cell invasion revealed by computer simulations. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1189-1202. doi: 10.3934/mbe.2015.12.1189 [18] M. Dolfin, D. Knopoff, L. Leonida, D. Maimone Ansaldo Patti. Escaping the trap of 'blocking': A kinetic model linking economic development and political competition. Kinetic & Related Models, 2017, 10 (2) : 423-443. doi: 10.3934/krm.2017016 [19] Yang Woo Shin, Dug Hee Moon. Throughput of flow lines with unreliable parallel-machine workstations and blocking. Journal of Industrial & Management Optimization, 2017, 13 (2) : 901-916. doi: 10.3934/jimo.2016052 [20] Thomas Dauer, Marlies Gerber. Generic absence of finite blocking for interior points of Birkhoff billiards. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4871-4893. doi: 10.3934/dcds.2016010

2020 Impact Factor: 0.935