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On the parameters of extended primitive cyclic codes and the related designs

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Abstract

Very recently, Heng et al. studied a family of extended primitive cyclic codes. It was shown that the supports of all codewords with any fixed nonzero Hamming weight in this code support a 2-design. In this paper, we study this family of extended primitive cyclic codes in more details. The weight distribution is determined and the parameters of the related 2-designs are also given. Moreover, we prove that the minimum weight codewords in this code support a 3-design when \(p=2\), which gives an affirmative answer to Heng’s conjecture.

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Acknowledgements

The authors would like to thank the Associate Editor and the anonymous Reviewers for giving us invaluable comments and suggestions that greatly improved the quality of this paper. This work was supported by the Fundamental Research Funds for the Central Universities of China (Grant No. 2682023ZTPY002) and the National Natural Science Foundation of China (Grant No. 12231015).

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  1. School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China

    Haode Yan & Yanan Yin

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  1. Haode Yan
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  2. Yanan Yin
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Correspondence to Haode Yan.

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Yan, H., Yin, Y. On the parameters of extended primitive cyclic codes and the related designs. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s10623-024-01357-2

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