Primal-dual distance bounds of linear codes with application to cryptography
<|reference_start|>Primal-dual distance bounds of linear codes with application to cryptography: Let $N(d,d^\perp)$ denote the minimum length $n$ of a linear code $C$ with $d$ and $d^{\bot}$, where $d$ is the minimum Hamming distance of $C$ and $d^{\bot}$ is the minimum Hamming distance of $C^{\bot}$. In this paper, we show a lower bound and an upper bound on $N(d,d^\perp)$. Further, for small values of $d$ and $d^\perp$, we determine $N(d,d^\perp)$ and give a generator matrix of the optimum linear code. This problem is directly related to the design method of cryptographic Boolean functions suggested by Kurosawa et al.<|reference_end|>
@article{matsumoto2005primal-dual,
    title={Primal-dual distance bounds of linear codes with application to
  cryptography},
    author={Ryutaroh Matsumoto, Kaoru Kurosawa, Toshiya Itoh, Toshimitsu Konno,
  Tomohiko Uyematsu},
    journal={IEEE Trans. Inform. Theory, vol. 52, no. 9, pp. 4251-4256, Sept.
  2006},
    year={2005},
    doi={10.1109/TIT.2006.880050},
    archivePrefix={arXiv},
    eprint={cs/0506087},
    primaryClass={cs.IT cs.CR math.IT}
}