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Some optimal cyclic Fq-linear Fqt-codes

  • * Corresponding author: Yun Gao

    * Corresponding author: Yun Gao 

The authors would like to thank the anonymous reviewers and the Associate Editor for their valuable suggestions and comments that helped to greatly improve the paper. This research is supported by the 973 Program of China (Grant No. 2013CB834204), the National Natural Science Foundation of China (Grant No. 11671024, 61571243), and the Fundamental Research Funds for the Central Universities of China

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  • Let Fqt be a finite field of cardinality qt, where q is a power of a prime number p and t1 is a positive integer. Firstly, a family of cyclic Fq-linear Fqt-codes of length n is given, where n is a positive integer coprime to q. Then according to the structure of this kind of codes, we construct 60 optimal cyclic Fq-linear Fq2-codes which have the same parameters as the MDS codes over Fq2.

    Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 11T71.

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  • Table 1.  Some optimal cyclic Fq-linear Fq2-codes of length n

    {q,n} Basis Hamming weight enumerator (n,(q2)k,d)
    {2,5} α1 W1 (5,(22)2,4)
    {3,5} α2 W2 (5,(32)2,4)
    {3,7} α3 W3 (7,(32)3,5)
    {5,7} α4 W4 (7,(52)3,5)
    {5,13} α5 W5 (13,(52)2,12)
    {5,17} α6 W6 (17,(52)8,10)
    {7,5} α7 W7 (5,(72)2,4)
    {7,11} α8 W8 (11,(72)5,7)
    {7,13} α9 W9 (13,(72)6,8)
    {11,13} α10 W10 (13,(112)6,8)
    {11,17} α11 W11 (17,(112)8,10)
    {13,5} α12 W12 (5,(132)2,4)
    {13,11} α13 W13 (11,(132)5,7)
    {13,17} α14 W14 (17,(132)2,16)
    {13,19} α15 W15 (19,(132)9,11)
    {17,5} α16 W16 (5,(172)2,4)
    {17,7} α17 W17 (7,(172)3,5)
    {17,11} α18 W18 (11,(172)5,7)
    {17,13} α19 W19 (13,(172)3,11)
    {19,7} α20 W20 (7,(192)3,5)
    {19,11} α21 W21 (11,(192)5,7)
    {19,13} α22 W22 (13,(192)6,8)
    {19,17} α23 W23 (17,(192)4,14)
    {23,5} α24 W24 (5,(232)2,4)
    {23,13} α25 W25 (13,(232)3,11)
    {23,17} α26 W26 (17,(232)8,10)
    {29,11} α27 W27 (11,(292)5,7)
    {29,17} α28 W28 (17,(292)8,10)
    {31,7} α29 W29 (7,(312)3,5)
    {31,13} α30 W30 (13,(312)2,12)
    {31,17} α31 W31 (17,(312)8,10)
    {37,5} α32 W32 (5,(372)2,4)
    {37,13} α33 W33 (13,(372)6,8)
    {41,11} α34 W34 (11,(412)5,7)
    {41,13} α35 W35 (13,(412)6,8)
    {43,5} α36 W36 (5,(432)2,4)
    {43,13} α37 W37 (13,(432)3,11)
    {43,17} α38 W38 (17,(432)4,14)
    {47,5} α39 W39 (5,(472)2,4)
    {47,7} α40 W40 (7,(472)3,5)
    {47,13} α41 W41 (13,(472)2,12)
    {47,17} α42 W42 (17,(472)2,16)
    {53,5} α43 W43 (5,(532)2,4)
    {53,17} α44 W44 (17,(532)4,14)
    {59,7} α45 W45 (7,(592)3,5)
    {59,13} α46 W46 (13,(592)6,8)
    {59,17} α47 W47 (17,(592)4,14)
    {61,7} α48 W48 (7,(612)3,5)
    {61,11} α49 W49 (11,(612)5,7)
    {67,5} α50 W50 (5,(672)2,4)
    {67,13} α51 W51 (13,(612)6,8)
    {71,13} α52 W52 (13,(712)6,8)
    {73,11} α53 W53 (11,(732)5,7)
    {73,13} α54 W54 (13,(732)2,12)
    {79,11} α55 W55 (11,(792)5,7)
    {83,5} α56 W56 (5,(832)2,4)
    {83,11} α57 W57 (11,(832)5,7)
    {89,7} α58 W58 (7,(892)3,5)
    {89,17} α59 W59 (17,(892)2,16)
    {97,5} α60 W60 (5,(972)2,4)
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