# 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
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##### References:
 [1] S. Ball, Multiple blocking sets and arcs in finite planes,, J. London Math. Soc. (2), 54 (1996), 581. Google Scholar [2] A. Betten and D. Betten, There is no Drake/Larson linear space on $30$ points,, J. Combin. Des., 18 (2010), 48. 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. doi: 10.1109/18.59932. Google Scholar [4] J. W. P. Hirschfeld, "Projective Geometries over Finite Fields,'' 2nd edition,, The Clarendon Press, (1998). Google Scholar [5] S. M. Johnson, A new upper bound for error-correcting codes,, IRE Trans., IT-8 (1962), 203. Google Scholar [6] F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes. II,'', North-Holland Publishing Co., (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. Google Scholar
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