Galois Hulls of Cyclic Codes over Finite Fields

Yang DING; Xiaohua LU

doi:10.1587/transfun.2019EAL2087

Galois Hulls of Cyclic Codes over Finite Fields

Yang DING, Xiaohua LU

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The Galois hull of linear code is defined to be the intersection of the code and its Galois dual. In this paper, we investigate the Galois hulls of cyclic codes over F_q^r. For any integer s≤r, we present some sufficient and necessary conditions that cyclic codes have l-dimensional s-Galois hull. Moreover, we prove that a cyclic code C has l-dimensional s-Galois hull iff C has l-dimensional (r-s)-Galois hull. In particular, we also present the sufficient and necessary condition for cyclic codes with 1-dimensional Galois hulls and the relationship between cyclic codes with 1-dimensional Galois hulls and cyclic codes with Galois complementary duals. Some optimal cyclic codes with Galois hulls are obtained. Finally, we explicitly construct a class of cyclic codes with 1-Galois linear complementary dual over F_q³.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E103-A No.1 pp.370-375

Publication Date: 2020/01/01

Publicized

Online ISSN: 1745-1337

DOI: 10.1587/transfun.2019EAL2087

Type of Manuscript: LETTER

Category: Coding Theory

Authors

Yang DING
Shanghai University
Xiaohua LU
Shanghai University

Keyword

s-Galois hull, cyclic codes, cyclotomic cosets, BCH bound

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Yang DING, Xiaohua LU, "Galois Hulls of Cyclic Codes over Finite Fields" in IEICE TRANSACTIONS on Fundamentals, vol. E103-A, no. 1, pp. 370-375, January 2020, doi: 10.1587/transfun.2019EAL2087.
Abstract: The Galois hull of linear code is defined to be the intersection of the code and its Galois dual. In this paper, we investigate the Galois hulls of cyclic codes over F_q^r. For any integer s≤r, we present some sufficient and necessary conditions that cyclic codes have l-dimensional s-Galois hull. Moreover, we prove that a cyclic code C has l-dimensional s-Galois hull iff C has l-dimensional (r-s)-Galois hull. In particular, we also present the sufficient and necessary condition for cyclic codes with 1-dimensional Galois hulls and the relationship between cyclic codes with 1-dimensional Galois hulls and cyclic codes with Galois complementary duals. Some optimal cyclic codes with Galois hulls are obtained. Finally, we explicitly construct a class of cyclic codes with 1-Galois linear complementary dual over F_q³.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2019EAL2087/_p

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@ARTICLE{e103-a_1_370,
author={Yang DING, Xiaohua LU, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Galois Hulls of Cyclic Codes over Finite Fields},
year={2020},
volume={E103-A},
number={1},
pages={370-375},
abstract={The Galois hull of linear code is defined to be the intersection of the code and its Galois dual. In this paper, we investigate the Galois hulls of cyclic codes over F_q^r. For any integer s≤r, we present some sufficient and necessary conditions that cyclic codes have l-dimensional s-Galois hull. Moreover, we prove that a cyclic code C has l-dimensional s-Galois hull iff C has l-dimensional (r-s)-Galois hull. In particular, we also present the sufficient and necessary condition for cyclic codes with 1-dimensional Galois hulls and the relationship between cyclic codes with 1-dimensional Galois hulls and cyclic codes with Galois complementary duals. Some optimal cyclic codes with Galois hulls are obtained. Finally, we explicitly construct a class of cyclic codes with 1-Galois linear complementary dual over F_q³.},
keywords={},
doi={10.1587/transfun.2019EAL2087},
ISSN={1745-1337},
month={January},}

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TY - JOUR
TI - Galois Hulls of Cyclic Codes over Finite Fields
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 370
EP - 375
AU - Yang DING
AU - Xiaohua LU
PY - 2020
DO - 10.1587/transfun.2019EAL2087
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E103-A
IS - 1
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - January 2020
AB - The Galois hull of linear code is defined to be the intersection of the code and its Galois dual. In this paper, we investigate the Galois hulls of cyclic codes over F_q^r. For any integer s≤r, we present some sufficient and necessary conditions that cyclic codes have l-dimensional s-Galois hull. Moreover, we prove that a cyclic code C has l-dimensional s-Galois hull iff C has l-dimensional (r-s)-Galois hull. In particular, we also present the sufficient and necessary condition for cyclic codes with 1-dimensional Galois hulls and the relationship between cyclic codes with 1-dimensional Galois hulls and cyclic codes with Galois complementary duals. Some optimal cyclic codes with Galois hulls are obtained. Finally, we explicitly construct a class of cyclic codes with 1-Galois linear complementary dual over F_q³.
ER -