The classification of (42,6)_8 arcs

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

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

Anton Betten ^1, , Eun Ju Cheon ^2, , Seon Jeong Kim ^2, and Tatsuya Maruta ^3,

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

Abstract
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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 $31$.

Keywords: projective plane., blocking set, Arc.

Mathematics Subject Classification: Primary: 51E21; Secondary: 05E1.

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:

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[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.
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References:

[1]	S. Ball, Multiple blocking sets and arcs in finite planes, J. London Math. Soc. (2), 54 (1996), 581-593.
[2]	A. Betten and D. Betten, There is no Drake/Larson linear space on $30$ points, J. Combin. Des., 18 (2010), 48-70.
[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.
[4]	J. W. P. Hirschfeld, "Projective Geometries over Finite Fields,'' 2^nd edition, The Clarendon Press, Oxford University Press, New York, 1998.
[5]	S. M. Johnson, A new upper bound for error-correcting codes, IRE Trans., IT-8 (1962), 203-207.
[6]	F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes. II,'' North-Holland Publishing Co., Amsterdam, 1977.
[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.

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