Homologous codes for multiple access channels

Abstract

Building on recent development by Padakandla and Pradhan, and by Lim, Feng, Pastore, Nazer, and Gastpar, this paper studies the potential of structured nested coset coding as a complete replacement for random coding in network information theory. The roles of two techniques used in nested coset coding to generate nonuniform codewords, namely, shaping and channel transformation, are clarified and illustrated via the simple example of the two-sender multiple access channel. While individually deficient, the optimal combination of shaping and channel transformation is shown to achieve the same performance as traditional random codes for the general two-sender multiple access channel. The achievability proof of the capacity region is extended to the multiple access channels with more than two senders, and with one or more receivers. A quantization argument consistent with the construction of nested coset codes is presented to prove achievability for their Gaussian counterparts. These results open up new possibilities of utilizing nested coset codes with the same generator matrix for a broader class of applications. I. INTRODUCTIONRandom independently and identically distributed (i.i.d.) code ensembles play a fundamental role in network information theory, with most existing coding schemes built on them; see, for example, [1]- [3]. As shown by the classical example by Körner and Marton [4], however, using the same code at multiple users can achieve strictly better performance for some communication problems. Recent studies illustrate the benefit of such structured coding for computing linear combinations in [5]-[10], for the interference channels in [11]-[14], and for the multiple access channels with state information in [15]. Consequently, there has been a flurry of research activities on structured coding in network information theory, facilitated in part by several standalone workshops and tutorials at major conferences by leading researchers.Most of the existing results are based on lattice codes or linear codes on finite alphabets. Recently, Padakandla and Pradhan [15] brought a new dimension to the arsenal of structured coding by developing nested coset codes for network information theory; see also Miyake [16] for nested coset codes for point-to-point communication. In these nested coset coding schemes, a coset code of a rate higher than the target is first generated randomly. A codeword of a desired property (such as type or joint type) is then selected from a subset (a coset of a subcode). This construction is reminiscent of the multicoding scheme in Gelfand-Pinsker coding for channels with state and Marton coding for broadcast channels. But in a sense, nested coset coding is more fundamental in that the scheme at its core is relevant even for single-user communication. By a careful combination of individual and common parts of coset codes, the proposed coding scheme in [15] achieves rates for multiple access channels (MACs) with state beyond what can be achieved by existing random or structured coding sche...

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“…A similar phenomenon occurs for the nested linear coding region for the multiple-access channel. However, recent work by Sen and Kim [12] has shown that, for any rate pair in the multipleaccess capacity region, one can always select a finite field and pmf such that this rate pair is included in the nested linear coding region. An interesting question for future study is whether the same is true for distributed source coding.…”

“…We are now ready to proceed with the last probability term P (E 3 ∩ E c 1 |M) using the union of events bound. Continuing from (12),…”

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“…One by-product of our analysis is an achievable rate region for multiple access via nested linear codes. Recent work [47] has further explored this rate region and shown, through a careful selection of both the finite field and symbol mappings, that it in fact corresponds to the full multiple-access capacity region.…”

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“…For multiple encoders, the desired common structure is obtained by using coset codes with the same generator matrix. Recent efforts exploited the benefit of such constructions for a broader class of channel models, such as interference channels [17], [18], multiple access channels [19], [20], and multiple access channels with state [21].…”

“…To develop a unified framework for the compute-forward strategy, Lim, Feng, Pastore, Nazer, and Gastpar [22], [23] generalized the nested coset codes of the same generator matrix to asymmetric rate pairs. We referred to this generalized version, together with the shaping step, as homologous codes [19], [20], [24]. This terminology is motivated from its biological definition, i.e., the structures modified from the same ancestry (underlying linear code) to adapt to different purposes (desired shape).…”