[论文解读] Low-Complexity Decoding for Symmetric, Neighboring and Consecutive Side-information Index Coding Problems

Mahesh Babu Vaddi, B. Sundar Rajan|arXiv (Cornell University)|May 9, 2017

Cooperative Communication and Network Coding参考文献 4被引用 12

一句话总结

本文提出了一种基于相邻独立行（AIR）矩阵的低复杂度解码方案，用于对称、相邻及连续的信道侧信息索引编码问题。通过利用AIR矩阵的组合性质，该方法通过识别每个接收端所需的最小广播符号与侧信息集合，降低了解码复杂度，使解码过程可仅通过简单的XOR运算完成，而非求解线性方程组，从而在减少域大小和计算成本的同时实现最优速率。

ABSTRACT

The capacity of symmetric, neighboring and consecutive side-information single unicast index coding problems (SNC-SUICP) with number of messages equal to the number of receivers was given by Maleki, Cadambe and Jafar. For these index coding problems, an optimal index code construction by using Vandermonde matrices was proposed. This construction requires all the side-information at the receivers to decode their wanted messages and also requires large field size. In an earlier work, we constructed binary matrices of size $m imes n (m\geq n)$ such that any $n$ adjacent rows of the matrix are linearly independent over every field. Calling these matrices as Adjacent Independent Row (AIR) matrices using which we gave an optimal scalar linear index code for the one-sided SNC-SUICP for any given number of messages and one-sided side-information. By using Vandermonde matrices or AIR matrices, every receiver needs to solve $K-D$ equations with $K-D$ unknowns to obtain its wanted message, where $K$ is the number of messages and $D$ is the size of the side-information. In this paper, we analyze some of the combinatorial properties of the AIR matrices. By using these properties, we present a low-complexity decoding which helps to identify a reduced set of side-information for each users with which the decoding can be carried out. By this method every receiver is able to decode its wanted message symbol by simply adding some index code symbols (broadcast symbols). We explicitly give both the reduced set of side-information and the broadcast messages to be used by each receiver to decode its wanted message. For a given pair or receivers our decoding identifies which one will perform better than the other when the broadcast channel is noisy.

研究动机与目标

• 降低对称、相邻及连续信道侧信息索引编码问题（SNC-SUICP）中的解码复杂度。
• 通过识别每个接收端所需的最小侧信息集合，消除解码过程中求解大规模线性方程组的需要。
• 利用AIR矩阵实现最优标量线性索引编码，同时减少域大小。
• 通过仅使用简单的XOR运算而非完整线性系统求解，实现高效解码。

提出的方法

• 该方法使用相邻独立行（AIR）矩阵，其任意n个相邻行在任意域上均保持线性无关。
• 分析AIR矩阵的组合性质，以识别每个接收端解码所需的最小广播符号与侧信息集合。
• 重构解码过程，使每个接收端通过XOR其指定的少量广播符号与已知的侧信息，即可解码其所需消息。
• 该方案利用消息依赖关系中的结构模式，通过选择性地添加广播符号来抵消干扰符号。
• 推导出每个接收端根据其在消息序列中的位置，所需广播符号与侧信息集合的显式表达式。
• 通过保持与先前基于Vandermonde构造相同的码长，确保实现最优速率，同时显著降低解码复杂度。

实验结果

研究问题

• RQ1 能否通过识别每个接收端所需的最小广播符号与侧信息集合，来降低SNC-SUICP中的解码复杂度？
• RQ2 如何利用AIR矩阵的组合结构，使解码过程简化至无需求解K−D个方程？
• RQ3 何种最小广播符号与侧信息集合可仅通过XOR运算实现成功解码？
• RQ4 与基于Vandermonde的构造相比，该方法是否在降低域大小与计算成本的同时保持最优速率？
• RQ5 在噪声广播信道中，该方案下各接收端的解码性能如何比较？

主要发现

• 所提出的解码方法通过用K−D个未知数的XOR运算替代求解K−D个方程，显著降低了复杂度。
• 每个接收端仅需使用一个显式定义的小型广播符号与侧信息子集，即可解码其所需消息，大幅减少计算负载。
• 该方法在保持与先前基于Vandermonde构造相同码长的前提下，实现了最优速率，同时显著降低了域大小需求。
• 对于位于区间[K−D−λ_l : K−D−1]的接收端，解码仅需XOR一个广播符号与已知侧信息，实现一步解码。
• 该方案支持在噪声信道中对不同接收端的性能进行比较，可基于所选广播符号识别出哪个接收端解码更可靠。
• 由于AIR矩阵的域无关线性无关性，该方法在任意有限域上均具有普适性，且不依赖于消息数K。