# Source Symbol Purging-Based Distributed Conditional Arithmetic Coding
## 1 Introduction  
Distributed Source Coding (DSC) is a non-traditional data compression architecture that independently encodes multiple sources and jointly decodes them. It is different from traditional multi-source joint coding schemes, which require encoders to communicate with each other for collaborative working. DSC requires two (or more) related sources to be independently encoded, and then jointly decoded. DSC is based on two important theorems: the first is the [PERSON] theorem [1], which proves that when data is losslessly compressed, independent coding and joint coding of related sources are equally effective; the second is the Wyner-Ziv theorem [2], which establishes the foundation for lossy DSC schemes. Today, DSC has led to achievements in many practical applications, such as: distributed video coding [3], hyperspectral image compression [4], multi-view video coding [5], wireless sensor network communication [6], image authentication [7], biometric information recognition [8] and so on. Considering the following application scenario when two correlated wireless sensor nodes (i.e., sources \(X\) and \(Y\)) sending data to a base station (the receiver), as long as the code rates \(R_{Y}\geqslant H(Y)\) and \(R_{X}\geqslant H(X|Y)\) are met after \(Y\) and \(X\) are encoded, lossless compression can be achieved. Obviously, source \(Y\) can be encoded using any traditional entropy coder. In the case that \(X\) and \(Y\) can communicate with each other, it is very simple for source \(X\) to be encoded at a rate \(R_{X}\geqslant H(X|Y)\) with an existing source coding scheme. However, communication between encoders consumes computing resources and the energy of the sensor nodes. Fortunately, the Slepian-Wolf theory guarantees that even no communication established between the two nodes, source \(X\) can also be encodedat a rate \(R(X)\) with \(H(X)>R_{X}\geqslant H(X|Y)\) and be losslessly decoded at the decoder although the encoder for source \(X\) knows nothing about the source \(Y\) (hence the distributed source coding). This is the asymmetric Slepian-Wolf problem that this work focuses on. In this scenario, source \(Y\) is regarded as the side information which is encoded using a traditional source code at a rate \(R_{Y}\geqslant H(Y)\). However, the coding scheme for source \(X\) is quite different, that is why the word 'asymmetric' is used. Therefore, designing a method that enables \(X\) to be encoded at a rate \(R(X)\) less than \(H(X)\) without knowing the characteristics of \(Y\) and how to use the correlation between \(X\) and \(Y\) to losslessly recover \(X\) during the decoding process are the research goals of asymmetric Slepian-Wolf coding [9; 10]. Apparently, the coding of \(Y\) is not the research focus of the asymmetric Slepian-Wolf problem; \(Y\) can usually be encoded by any lossless coding technology. Although the coding theory of DSC was proposed long ago, it was not until the 21 st century, when the important DSC scheme based on syndromes (DISCUS) was proposed for the asymmetric Slepian-Wolf problem in [11], that people carried out extensive research on specific coding schemes of DSC. After this, some DSC schemes based on channel coding have been proposed successively, such as the implementation of distributed source coding using Turbo codes [12], LDPC codes [13] and Polar codes [14]. The basic idea of DSC schemes based on channel codes is that the encoder only needs to send the syndromes or parity bits generated from the source sequence \(X\) to the decoder without the knowledge about \(Y\), and the number of bits required to transmit the syndromes will be less than the number of bits required to directly transmit \(X\) using traditional source codes, thereby achieving further compression of \(X\). The essence of the decoding process is that the decoder uses the received syndromes to 'correct' the side information sequence \(Y\) (the side information sequence \(Y\) can be regarded as an erroneous version of \(X\)).