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Error Correction for Index Coding With Side Information

机译：具有辅助信息的索引编码的纠错

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摘要

A problem of index coding with side information was first considered by Birk and Kol in 1998. In this study, a generalization of index coding scheme, where transmitted symbols are subject to errors, is studied. Error-correcting methods for such a scheme, and their parameters, are investigated. In particular, the following question is discussed: given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols $delta$ , what is the shortest length of a linear index code, such that every receiver is able to recover the required information? This question turns out to be a generalization of the problem of finding a shortest length error-correcting code with a prescribed error-correcting capability in the classical coding theory. The Singleton bound and two other bounds, referred to as the $alpha$-bound and the $kappa$ -bound, for the optimal length of a linear error-correcting index code (ECIC) are established. For large alphabets, a construction based on concatenation of an optimal index code with a maximum distance separable classical code is shown to attain the Singleton bound. For smaller alphabets, however, this construction may not be optimal. A random construction is also analyzed. It yields another inexplicit bound on the length of an optimal linear ECIC. Further, the problem of error-correcting decoding by a linear ECIC is studied. It is shown that in order to decode correctly the desired symbol, the decoder is required to find one of the vectors, belonging to an affine space containing the actual error vector. The syndrome decoding is shown to produce the correct output if the weight of the error pattern is less or equal to the error-correcting capability of the corresponding ECIC. Finally, the notion of static ECIC, whic- is suitable for use with a family of instances of an index coding problem, is introduced. Several bounds on the length of static ECICs are derived, and constructions for static ECICs are discussed. Connections of these codes to weakly resilient Boolean functions are established.
机译：Birk和Kol在1998年首先考虑了带有辅助信息的索引编码问题。在这项研究中，研究了索引编码方案的泛化，其中传输的符号容易出错。研究了这种方案的纠错方法及其参数。特别地，讨论以下问题：给定索引编码方案的边信息超图和最大错误符号$ delta $，线性索引码的最短长度是多少，以便每个接收器都能够恢复所需的信息？这个问题被证明是在经典编码理论中寻找具有规定的纠错能力的最短长度的纠错码的问题的概括。建立线性纠错索引代码（ECIC）最佳长度的Singleton边界和两个其他边界，分别称为$ alpha $绑定和$ kappa $绑定。对于大字母，示出了基于最佳索引代码与最大距离可分离古典代码的级联的构造，以达到单例边界。但是，对于较小的字母，此构造可能不是最佳的。还分析了随机构造。它在最优线性ECIC的长度上产生了另一个不确定的界限。此外，研究了通过线性ECIC进行的纠错解码的问题。示出了为了正确地解码期望的符号，要求解码器找到属于包含实际误差矢量的仿射空间的矢量之一。如果错误模式的权重小于或等于相应ECIC的错误纠正能力，则表明校正子解码会产生正确的输出。最后，介绍了静态ECIC的概念，该概念适合与一系列索引编码问题一起使用。推导了静态ECIC长度的几个界限，并讨论了静态ECIC的构造。这些代码与弱弹性布尔函数的连接已建立。

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