TY - JOUR
T1 - Extended cyclic codes, maximal arcs and ovoids
AU - Abdukhalikov, Kanat
AU - Ho, Duy
N1 - Funding Information:
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.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2021/10
Y1 - 2021/10
N2 - We show that extended cyclic codes over Fq 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 [q2+ 1 , 4 , q2- 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).
AB - We show that extended cyclic codes over Fq 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 [q2+ 1 , 4 , q2- 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).
KW - Extended cyclic codes
KW - Hyperovals
KW - Maximal arcs
KW - MDS codes
KW - Ovoids
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U2 - 10.1007/s10623-021-00915-2
DO - 10.1007/s10623-021-00915-2
M3 - Article
AN - SCOPUS:85111574676
VL - 89
SP - 2283
EP - 2294
JO - Designs, Codes, and Cryptography
JF - Designs, Codes, and Cryptography
SN - 0925-1022
IS - 10
ER -