We consider the design of coding schemes for the wireless two-way relaying channel when there is no channel state information at the transmitter. In the spirit of the compute-and-forward paradigm, we present a multilevel coding scheme that permits reliable computation (or, decoding) of a class of functions at the relay. The function to be computed (or decoded) is then chosen depending on the channel realization. We define such a class of functions which can be decoded at the relay using the proposed coding scheme and derive rates that are universally achievable over a set of channel gains when this class of functions is used at the relay. We develop our framework with general modulation formats in mind, but numerical results are presented for the case where each node transmits using 4-ary and 8-ary modulation schemes. Numerical results demonstrate that the flexibility afforded by our proposed scheme results in substantially higher rates than those achievable by always using a fixed function or considering only linear functions over higher order fields.

We study modulo-sum computation of two binary source sequences over a two-user erasure multiple access channel. The channel is modeled as a binary-input, erasure multiple access channel, which can be in one of three states-either the channel output is a modulo-sum of the two input symbols, or the channel output equals the input symbol on the first link and an erasure on the second link, or vice versa. The associated state sequence is independent and identically distributed. Unlike previously studied multiple-access channels, the proposed channel is not matched to the modulo-sum function and therefore we expect simple cut-set upper bounds to be far from capacity. In this paper, we establish a new upper bound on the modulo-sum capacity that is tighter than the cut-set bound. The key step in establishing this new bound is to provide suitable side information to the encoders to reduce the setup to a compound multiple-access channel and then capture the tension across multiple receivers required to compute the modulo-sum function. In our lower bound, it suffices to use identical linear codebooks at the two encoders. When a (strictly) causal feedback of the channel state is available to the encoders, we present a simple coding scheme that can achieve a rate larger than our upper bound for the case without feedback. This shows that the modulo-sum capacity is increased with feedback. An extension to the case of lossy reconstruction is also treated briefly.

We consider the design and analysis of coding schemes for the binary input two way relay channel with erasure noise. We focus on reliable physical layer network coding as described in [1] in which the relay performs perfect error correction prior to forwarding messages. The best known achievable rates for this problem can be achieved through either decode and forward or compute and forward relaying. We consider a decoding paradigm called joint compute and forward which we numerically show can achieve the best of these rates with a single encoder and decoder. This is accomplished by deriving the exact performance of a message passing decoder based on joint compute and forward for spatially coupled LDPC ensembles.

We study computation of a modulo-sum of two binary source sequences over a two-user erasure multiple access channel. Each sender observes an independent and equiprobable binary sequence and the receiver is interested in computing the modulo-sum of these two sequences. The channel is modelled as a binary-input, erasure multiple access channel, which can be in one of three states - either the channel output is a modulo-sum of the two input symbols, or the channel output equals the input symbol on the first link and an erasure on the second link, or it equals the input symbol on the second link and an erasure on the first link. The associated state sequence is independent and identically distributed. We establish upper and lower bounds on the modulo-sum capacity. Our coding scheme uses either the compute-and-forward or the decode-and-forward techniques. The upper bound is obtained by a genie aided argument that reduces the setup to a compound multiple-access channel. It is in general is tighter than a simple upper bound obtained by revealing one of the messages to the decoders. We also briefly consider the case when a strictly causal state feedback is available to the encoders and establish that such feedback can increase the modulo-sum capacity.

We consider the design of coding schemes for the wireless two-way relaying channel when there is no channel state information at the transmitter. In the spirit of the compute and forward paradigm, we present a multilevel coding scheme that permits the recovery of a class of functions at the relay. We define such a class of functions and derive rates that are universally achievable over a set of channel gains when this class of functions is used at the relay. We develop our framework with general modulation formats in mind, but numerical results are presented for the case where each node transmits using the QPSK constellation. Numerical results with QPSK show that substantially higher rates are achievable with our proposed approach than those achievable by always using a fixed function or adapting the function at the relay but coding over GF(4).