Abstract
We show that extended cyclic codes over \(\mathbb {F}_q\) with parameters \([q+2,3,q]\), \(q=2^m\), determine regular hyperovals. We also show that extended cyclic codes with parameters \([qt-q+t,3,qt-q]\), \(1<t<q\), q is a power of t, determine (cyclic) Denniston maximal arcs. Similarly, cyclic codes with parameters \([q^2+1,4,q^2-q]\) are equivalent to ovoid codes obtained from elliptic quadrics in PG(3, q). Finally, we give simple presentations of Denniston maximal arcs in PG(2, q) and elliptic quadrics in PG(3, q).
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Acknowledgements
The authors would like to thank Cunsheng Ding for valuable discussions and suggestions. The author is also grateful to the anonymous reviewers for their detailed comments that improved the presentation and quality of this paper. This work was supported by UAEU Grant 31S366 and grant AP09259551.
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Communicated by V. D. Tonchev.
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Abdukhalikov, K., Ho, D. Extended cyclic codes, maximal arcs and ovoids. Des. Codes Cryptogr. 89, 2283–2294 (2021). https://doi.org/10.1007/s10623-021-00915-2
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DOI: https://doi.org/10.1007/s10623-021-00915-2