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# Optimal binary linear codes from posets of the disjoint union of two chains

• *Corresponding author: Yansheng Wu

This work was supported by the National Natural Science Foundation of China (Nos. 12101326, 62172219), the Natural Science Foundation of Jiangsu Province (No. BK20210575), the Natural Science Research Project of Colleges and Universities in Jiangsu Province (No. 21KJB110005), the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST)(NRF-2017R1D1A1B05030707), and the Foundation of State Key Laboratory of Integrated Services Networks under Grant ISN23-22

• Recently, some infinite families of optimal binary linear codes are constructed from simplicial complexes. Afterwards, the construction method was extended to using arbitrary posets. In this paper, based on a generic construction of linear codes, we obtain four classes of optimal binary linear codes by using the posets of two chains. Two of them induce Griesmer codes which are not equivalent to the linear codes constructed by Belov. Those codes are exploited to construct secret sharing schemes in cryptography as well.

Mathematics Subject Classification: Primary: 94B05.

 Citation: • • Figure 1.  $\mathbb{P} = ([m,n], \leq)$

Table 1.  Theorem 3.1 (Ⅰ)

 Weight Frequency $2^{n-1}-i+s(0\le s< i)$ $2^{n-i}{i\choose s}$ $2^{n-1}$ $2^{n-i}-1$

Table 2.  Theorem 3.1 (Ⅱ)

 Weight Frequency $2^{n-1}-(j-m)+t(0\le t< j-m)$ $2^{n-(j-m)}{j-m\choose t}$ $2^{n-1}$ $2^{n-(j-m)}-1$

Table 3.  Theorem 3.1 (Ⅲ)

 Weight Frequency $\begin{array}{*{20}{c}} {{2^{n - 1}} + s + t + 2st - (s + 1)(j - m) - (t + 1)i}\\ {0 \le s \le i,0 \le t \le j - m,}\\ {(s,t) \ne (i,j - m)} \end{array}$ $2^{n-i-(j-m)}{i\choose s}{j-m\choose t}$ $2^{n-1}$ $2^{n-i-(j-m)}-1$
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Tables(3)

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