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Some bounds on binary LCD codes

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Abstract

A linear code with a complementary dual (or An LCD code) is defined to be a linear code C whose dual code C ⊥ satisfies C ∩ C ⊥= \(\left \{ \mathbf {0}\right \} \). Let L D (n, k) denote the maximum of possible values of d among [n, k, d] binary LCD codes. We give the exact values of L D (n, k) for k = 2 for all n and some bounds on L D (n, k) for other cases. From our results and some direct search we obtain a complete table for the exact values of L D (n, k) for 1 ≤ k ≤ n ≤ 12. As a consequence, we also derive bounds on the dimensions of LCD codes with fixed lengths and minimum distances.

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Acknowledgments

J.-L. Kim was supported by Basic Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03933259).

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Authors and Affiliations

  1. Department of Mathematics, Sogang University, Seoul, 04107, South Korea

    Lucky Galvez, Jon-Lark Kim, Nari Lee, Young Gun Roe & Byung-Sun Won

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  1. Lucky Galvez
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  2. Jon-Lark Kim
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  3. Nari Lee
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  4. Young Gun Roe
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  5. Byung-Sun Won
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Correspondence to Jon-Lark Kim.

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Galvez, L., Kim, JL., Lee, N. et al. Some bounds on binary LCD codes. Cryptogr. Commun. 10, 719–728 (2018). https://doi.org/10.1007/s12095-017-0258-1

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  • DOI: https://doi.org/10.1007/s12095-017-0258-1

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