How to achieve the capacity of asymmetric channels

Abstract

We survey coding techniques that enable reliable transmission at rates that approach the capacity of an arbitrary discrete memoryless channel. In particular, we take the point of view of modern coding theory and discuss how recent advances in coding for symmetric channels help provide more efficient solutions for the asymmetric case. We consider, in more detail, three basic coding paradigms.The first one is Gallager's scheme that consists of concatenating a linear code with a non-linear mapping so that the input distribution can be appropriately shaped. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Furthermore, we derive a scaling law between the gap to capacity, the cardinality of the input and output alphabets, and the required size of the mapper.The second one is an integrated scheme in which the code is used both for source coding, in order to create codewords distributed according to the capacity-achieving input distribution, and for channel coding, in order to provide error protection. Such a technique has been recently introduced by Honda and Yamamoto in the context of polar codes, and we show how to apply it also to the design of sparse graph codes.The third paradigm is based on an idea of Böcherer and Mathar, and separates the two tasks of source coding and channel coding by a chaining construction that binds together several codewords. We present conditions for the source code and the channel code, and we describe how to combine any source code with any channel code that fulfill those conditions, in order to provide capacity-achieving schemes for asymmetric channels. In particular, we show that polar codes, spatially coupled codes, and homophonic codes are suitable as basic building blocks of the proposed coding strategy.Rather than focusing on the exact details of the schemes, the purpose of this tutorial is to present different coding techniques that can then be implemented with many variants. There is no absolute winner and, in order to understand the most suitable technique for a specific application scenario, we provide a detailed comparison that takes into account several performance metrics.

Rate-Constrained Shaping Codes for Structured Sources

“…Gallager [16, p. 208] proposed a method of generating symbols with a biased distribution to be combined with linear coding as an approach to achieving capacity of an asymmetric channel. This idea was incorporated into a general scheme that can use capacity-achieving codes for symmetric channels, such as polar codes, to achieve the capacity of arbitrary discrete memoryless asymmetric channels in Mondelli et al [43].…”

“…This scheme can be regarded as a simplification of the bootstrap scheme in Böcherer and Mathar [7], which concatenates the check bits generated by the systematic ECC encoder with the following information bits and applies a DM encoder to them. In [43], the authors also proved that the bootstrap scheme, which they refer to as a chaining construction, can be used to achieve the capacity of any discrete memoryless asymmetric channel.…”

Shaped On–Off Keying Using Polar Codes

“…This method is called randomized rounding rule in [11]. A simplified approach is an encoding rule called the argmax rule in [11]. Here, for the values of u i with i ∈ D, one chooses…”

“…In this work, we analyze a probabilistic shaping (PS) approach for OOK that uses a method by Honda and Yamamoto [10], [11] where polar codes [12], [13] perform joint distribution matching and FEC. This idea was also applied in [14] with the intention to avoid an additional DM [15] and to use a single component for distribution matching and FEC.…”

Improved demapping for channels with data-dependent noise

“…For example, we could search for a channel coding scheme to achieve capacity using a nonuniform distribution of symbols [19]. Alternatively, the input symbols could be transformed by a nonlinear function to create a uniform noise distribution at the receiver, as was proposed for optical systems [16].…”