具有近最优性能的单形反码的高效多项式时间解码
Efficient Polynomial-Time Decoding of Simplicial Anticodes with Near-Optimal Performance
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中文总结 AI 辅助
针对单形复形生成的二元线性码，提出一种高效多项式时间解码算法，其纠错能力界渐近最优，在特定单形复形族上可达到理论界。
中文摘要 AI 辅助
在本研究中，我们针对由单形复形生成的码提出了一种高效解码算法，这类码是一类二元线性码，此前尚无此类解码方法被提出。尽管该算法并非总能达到理论上的最大纠错能力，但它提供了一个可直接从复形结构计算得到的显式界。此外，该界具有渐近最优性：在极大面维数的自然假设下，随着码长增加，算法保证的纠错能力与理论最大纠错能力的比值趋近于1。本文还在具体示例中给出了纠错能力的相关结果。最后，我们引入了特定的单形复形族，在这些复形上该算法可成功达到上述理论界。
英文摘要
In this work, we propose an efficient decoding algorithm for codes arising from simplicial complexes, a family of binary linear codes for which no decoding method of this type was previously known. Although the algorithm does not always attain the maximum theoretical error-correcting capability, it provides an explicit bound that can be computed directly from the structure of the complex. Moreover, this bound is asymptotically optimal: the ratio between the guaranteed correcting capability and the theoretical maximum converges to $1$ as the code length increases, under natural assumptions on the dimension of the maximal faces. The correction capability is also presented in specific examples. Finally, we introduce specific families of simplicial complexes where the algorithm successfully reaches this theoretical bound.
发表机构
- University of Almería（阿尔梅里亚大学）
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