一种基于子包化的嵌入式索引码构造方法
An Embedded Index Code Construction Using Sub-packetization
摘要：
索引编码问题（ICP）的一个变体，嵌入式索引编码问题（EICP）在[A. Porter和M. Wootters，“Embedded index coding，”ITW，Sweden，2019]，其动机是其在分布式计算中的应用，其中每个用户都可以充当其他用户的发送者，并且报告了用于代码构造的算法。其构造依赖于矩阵的最小秩的计算，这是计算密集型的。在[A. Mahesh，N. Sageer Karat和B. S. Rajan，“Min-rank of Embedded Index Coding Problems，”ISIT，2020]，对于EICP，引入了边信息矩阵的概念，并且证明了最佳标量线性索引码的长度等于边信息矩阵的最小秩。作者为一类EICP - \textit{Consecutive and Symmetric Embedded Index Coding Problem (CS-EICP)}提供了一个显式的代码构造。我们引入子分组化的消息的索引编码问题的想法，提供一个新的代码建设CS-EICP在以前的作品中提供的标量线性解决方案相比。对于CS-EICP，对于我们的构造，归一化速率（其被定义为由所有消息的比特总数归一化的所有用户一起发送的比特数）小于Mahesh等人实现的归一化速率，对于标量线性码。
英文摘要：
A variant of the index coding problem (ICP), the embedded index coding problem (EICP) was introduced in [A. Porter and M. Wootters, "Embedded index coding," ITW, Sweden, 2019] which was motivated by its application in distributed computing where every user can act as sender for other users and an algorithm for code construction was reported. The constructions depends on the computation of minrank of a matrix, which is computationally intensive. In [A. Mahesh, N. Sageer Karat and B. S. Rajan, "Min-rank of Embedded Index Coding Problems," ISIT, 2020], for EICP, a notion of side-information matrix was introduced and it was proved that the length of an optimal scalar linear index code is equal to the min-rank of the side-information matrix. The authors have provided an explicit code construction for a class of EICP - \textit{Consecutive and Symmetric Embedded Index Coding Problem (CS-EICP)}. We introduce the idea of sub-packetization of the messages in index coding problems to provide a novel code construction for CS-EICP in contrast to the scalar linear solutions provided in the prior works. For CS-EICP, the normalized rate, which is defined as the number of bits transmitted by all the users together normalized by the total number of bits of all the messages, for our construction is lesser than the normalized rate achieved by Mahesh et al.,for scalar linear codes.
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