Maximal arcs and extended cyclic codes

被引：6
作者：
De Winter, Stefaan [1 ]
Ding, Cunsheng [2 ]
Tonchev, Vladimir D. [3 ]
机构：
[1] Michigan Technol Univ, Houghton, MI 49921 USA
[2] Hong Kong Univ Sci & Technol, Hong Kong, Peoples R China
[3] Michigan Technol Univ, Houghton, MI 49931 USA
来源：
DESIGNS CODES AND CRYPTOGRAPHY | 2019年 / 87卷 / 04期
关键词：
Maximal arc; 2-Design; Two-weight code; Cyclic code; PROJECTIVE-PLANES; WEIGHTS;
D O I：
10.1007/s10623-018-0514-1
中图分类号：
TP301 [理论、方法];
学科分类号：
081202 ;
摘要：
It is proved that for every d2 such that d-1 divides q-1, where q is a power of 2, there exists a Denniston maximal arc A of degree d in PG(2,q), being invariant under a cyclic linear group that fixes one point of A and acts regularly on the set of the remaining points of A. Two alternative proofs are given, one geometric proof based on Abatangelo-Larato's characterization of Denniston arcs, and a second coding-theoretical proof based on cyclotomy and the link between maximal arcs and two-weight codes.
引用
收藏
页码：807 / 816
页数：10
相关论文
共 23 条
  • [1] ABATANGELO V, 1989, GEOMETRIAE DEDICATA, V30, P197
  • [2] Assmus E. F. Jr., 1969, Journal of Combinatorial Theory, Series A, V6, P122, DOI 10.1016/S0021-9800(69)80115-8
  • [3] Maximal arcs in Desarguesian planes of odd order do not exist
    Ball, S
    Blokhuis, A
    Mazzocca, F
    [J]. COMBINATORICA, 1997, 17 (01) : 31 - 41
  • [4] Barlotti A., 1956, BOLL MAT ITAL 3, V11, P553
  • [5] WEIGHTS OF IRREDUCIBLE CYCLIC CODES
    BAUMERT, LD
    MCELIECE, RJ
    [J]. INFORMATION AND CONTROL, 1972, 20 (02): : 158 - &
  • [6] THE GEOMETRY OF 2-WEIGHT CODES
    CALDERBANK, R
    KANTOR, WM
    [J]. BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 1986, 18 : 97 - 122
  • [7] A geometric approach to Mathon maximal arcs
    De Clerck, F.
    De Winter, S.
    Maes, T.
    [J]. JOURNAL OF COMBINATORIAL THEORY SERIES A, 2011, 118 (04) : 1196 - 1211
  • [8] Delsarte P., 1971, R160 MBLE RES LAB, VR160
  • [9] Denniston R., 1969, J. Comb. Theory, V6, P317
  • [10] Hamming weights in irreducible cyclic codes
    Ding, Cunsheng
    Yang, Jing
    [J]. DISCRETE MATHEMATICS, 2013, 313 (04) : 434 - 446
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