Galois Hulls of Cyclic Codes over Finite Fields

Yang DING  Xiaohua LU  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E103-A   No.1   pp.370-375
Publication Date: 2020/01/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.2019EAL2087
Type of Manuscript: LETTER
Category: Coding Theory
Keyword: 
s-Galois hull,  cyclic codes,  cyclotomic cosets,  BCH bound,  

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Summary: 
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 Fqr. For any integer sr, 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 Fq3.