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A class of binary cyclic codes and sequence families

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Abstract

For two odd integers l, k with \(0<l<k\) and \(\gcd (l,k)=1\), let \(m=2k\) and \(d=\frac{2^{lk}+1}{2^l+1}+\frac{2(2^m-1)}{3}\). In this paper, we determine the value distribution of the exponential sum \(\sum _{x\in \mathbb {F}_{2^m}}(-1)^{\mathrm {Tr}_1^m(ax+bx^d)}\). As applications, the weight distribution of a class of binary cyclic codes is settled. Second, we determine the correlation distribution among sequences in a sequence family.

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Acknowledgments

The authors would like to thank the anonymous referees for their helpful comments that improved the presentation of the paper. The work of H. Liang was supported by the National Natural Science Foundation of China under Grant 11201169 and the Postgraduate Innovation Project of Jiangsu Province under Grant KYZZ15\(_{-}\)0360. The work of Y. Tang was supported by the National Natural Science Foundation of China under Grant 61379004.

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Authors and Affiliations

  1. School of Mathematical Sciences, Yangzhou University, Yangzhou, 225002, China

    Hua Liang & Yuansheng Tang

  2. School of Mathematical Sciences, Huaiyin Normal University, Huaian, 223300, China

    Hua Liang

  3. School of Mathematics and Computation Science, Anqing Normal University, Anqing, 246133, China

    Wenbing Chen

Authors
  1. Hua Liang
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  2. Wenbing Chen
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  3. Yuansheng Tang
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Correspondence to Hua Liang.

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Liang, H., Chen, W. & Tang, Y. A class of binary cyclic codes and sequence families. J. Appl. Math. Comput. 53, 733–746 (2017). https://doi.org/10.1007/s12190-016-0993-z

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  • DOI: https://doi.org/10.1007/s12190-016-0993-z

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