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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

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
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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

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