Construction of Two Classes of Minimal Binary Linear Codes
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摘要: 线性码在数据存储、信息安全以及秘密共享等领域具有重要的作用。而极小线性码是设计秘密共享方案的首选码，设计极小线性码是当前密码与编码研究的重要内容之一。该文首先选取恰当的布尔函数，研究了函数的Walsh谱值分布，并利用布尔函数的Walsh谱值分布构造了两类极小线性码，确定了码的参数及重量分布。结果表明，所构造的码是不满足Ashikhmin-Barg条件的极小线性码，可用作设计具有良好访问结构的秘密共享方案。Abstract: Linear codes play an important role in data storage, information security and secret sharing. Minimal linear codes are the first choice to design secret sharing schemes, so the design of minimal linear codes is one of the important contents of current cryptosystem and coding theory. In this paper, the Walsh spectrum distribution of the selected Boolean functions is studied, and two kinds of minimal linear codes are obtained by using the Walsh spectrum distribution of the functions, then the weight distribution of the codes are determined. The results show that the constructed codes are minimal linear codes that do not satisfy the Ashikhmin-Barg condition, and can be used to design secret sharing schemes with good access structure.
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Key words:
- Boolean functions /
- Walsh transform /
- Binary linear codes
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表 1 码
${{\boldsymbol{C}}_{\boldsymbol{f}}}$ 的重量分布重量 频数 $ 0 $ $ 1 $ $ {2^{m - 1}} $ ${2^m} - 1 + {2^s}\left({2^t} - 2 - \dfrac{ {s(s + 1)} }{2}\right) + {\varepsilon _1}\left( {\begin{array}{*{20}{c} } s \\ {(2s + 3 \pm k)/4} \end{array} } \right)$ $ {2^{m - 1}} - {2^{t - 1}}A(i) $ $\begin{array}{*{20}{c} } {\left( {\begin{array}{*{20}{c} } s \\ i \end{array} } \right)}&{\left(1 \le i \le s,i \ne \dfrac{ {2s + 3 \pm k} }{4}\right)} \end{array}$ $ {2^{m - 1}} - {2^{t - 1}} $ $ {2^{s - {\text{2}}}}s(s + 1) + {\varepsilon _2}\left( {\begin{array}{*{20}{c}} s \\ {(2s + 3 \pm k)/4} \end{array}} \right) $ $ {2^{m - 1}} + {2^{t - 1}} $ $ {2^s} + {2^{s - {\text{2}}}}s(s + 1) + {\varepsilon _3}\left( {\begin{array}{*{20}{c}} s \\ {(2s + 3 \pm k)/4} \end{array}} \right) $ ${2^{m - 1} } + {2^{t - 1} }\left({2^s} - 1 - \dfrac{ {s(s + 1)} }{2}\right)$ $ 1 $ 
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表 2 码
${{\boldsymbol{C}}_{{{\boldsymbol{f}}_{\bar {\boldsymbol{D}}}}}}$ 的重量分布重量 频数 $ 0 $ $ 1 $ $ {2^{m - 1}} $ $ {2^m} - 1 $ $ {2^{m - 1}} + {2^{t - 1}}A(i) - 1 $ $\begin{array}{*{20}{c} } {\left( {\begin{array}{*{20}{c} } s \\ i \end{array} } \right)}&{ \left(1 \le i \le s,i \ne \dfrac{ {2s + 3 \pm k} }{4}\right)} \end{array}$ $ {2^{m - 1}} - 1 $ ${2^s}\left({2^t} - 2 - \dfrac{ {s(s + 1)} }{2}\right) + {\varepsilon _1}\left( {\begin{array}{*{20}{c} } s \\ {(2s + 3 \pm k)/4} \end{array} } \right)$ $ {2^{m - 1}} + {2^{t - 1}} - 1 $ $ {2^{s - {\text{2}}}}s(s + 1) + {\varepsilon _3}\left( {\begin{array}{*{20}{c}} s \\ {(2s + 3 \pm k)/4} \end{array}} \right) $ $ {2^{m - 1}} - {2^{t - 1}} - 1 $ $ {2^s} + {2^{s - {\text{2}}}}s(s + 1) + {\varepsilon _2}\left( {\begin{array}{*{20}{c}} s \\ {(2s + 3 \pm k)/4} \end{array}} \right) $ ${2^{m - 1} } - {2^{t - 1} }\left({2^s} - 1 - \dfrac{ {s(s + 1)} }{2}\right) - 1$ $ 1 $
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[1] ÇALKAVUR S. A study on multisecret-sharing schemes based on linear codes[J]. Emerging Science Journal, 2020, 4(4): 263–271. doi: 10.28991/esj-2020-01229 [2] WANG Yaru, LI Fulin, and ZHU Shixin. Two-weight linear codes and their applications in secret sharing[J]. Chinese Journal of Electronics, 2019, 28(4): 706–711. doi: 10.1049/cje.2019.04.006 [3] CHABANNE H, COHEN G, and PATEY A. Towards secure two-party computation from the wire-tap channel[C]. The 16th International Conference on Information Security and Cryptology, Seoul, Korea, 2014: 34–46. [4] NIEMINEN R and JÄRVINEN K. Practical privacy-preserving indoor localization based on secure two-party computation[J]. IEEE Transactions on Mobile Computing, 2021, 20(9): 2877–2890. doi: 10.1109/TMC.2020.2990871 [5] WANG Qichun and STĂNICĂ P. New bounds on the covering radius of the second order Reed-Muller code of length 128[J]. Cryptography and Communications, 2019, 11(2): 269–277. doi: 10.1007/s12095-018-0289-2 [6] CARLET C. The automorphism groups of the Kerdock codes[J]. Journal of Information and Optimization Sciences, 1991, 12(3): 387–400. doi: 10.1080/02522667.1991.10699078 [7] BAUMERT L D and MCELIECE R J. Weights of irreducible cyclic codes[J]. Information and Control, 1972, 20(2): 158–175. doi: 10.1016/S0019-9958(72)90354-3 [8] DING Cunsheng and NIEDERREITER H. Cyclotomic linear codes of order 3[J]. IEEE Transactions on Information Theory, 2007, 53(6): 2274–2277. doi: 10.1109/TIT.2007.896886 [9] XIANG Can. Linear codes from a generic construction[J]. Cryptography and Communications, 2016, 8(4): 525–539. doi: 10.1007/s12095-015-0158-1 [10] DING Cunsheng. A construction of binary linear codes from Boolean functions[J]. Discrete Mathematics, 2016, 339(15): 2288–2303. doi: 10.1016/j.disc.2016.03.029 [11] CHANG S and HYUN J Y. Linear codes from simplicial complexes[J]. Designs, Codes and Cryptography, 2018, 86(10): 2167–2181. doi: 10.1007/s10623-017-0442-5 [12] HENG Ziling, DING Cunsheng, and ZHOU Zhengchun. Minimal linear codes over finite fields[J]. Finite Fields and Their Applications, 2018, 54: 176–196. doi: 10.1016/j.ffa.2018.08.010 [13] DING Cunsheng, HENG Ziling, and ZHOU Zhengchun. Minimal binary linear codes[J]. IEEE Transactions on Information Theory, 2018, 64(10): 6536–6545. doi: 10.1109/TIT.2018.2819196 [14] MESNAGER S, QI Yanfeng, RU Hongming, et al. Minimal linear codes from characteristic functions[J]. IEEE Transactions on Information Theory, 2020, 66(9): 5404–5413. doi: 10.1109/TIT.2020.2978387 [15] ASHIKHMIN A and BARG A. Minimal vectors in linear codes[J]. IEEE Transactions on Information Theory, 1998, 44(5): 2010–2017. doi: 10.1109/18.705584 [16] MACWILLIAMS F J and SLOANE N J A. The Theory of Error-Correcting Codes[M]. Amsterdam: Elsevier-North-Holland, 1997. -
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