This paper deals with polarization-multiplexed optical transmissions. It addresses the challenge of designing optimal yet practical modulation codes for the 2×2 MIMO fiber channel. This specific use case diverges from classical MIMO models encountered, e.g., in the wireless literature as, here, non-ergodicity and low-complexity are key. While rather simple from a fundamental viewpoint, the proposed modulation schemes appears to be optimal and geometrically pleasing as the second minimum distance turns out to be the object of interest and the associated vectors encode the basis of a regular tetrahedron in the affine space. From an operational viewpoint, these new schemes for single channel use form polarization codes that achieve unit dB gains while exhibiting low implementation complexity.

We generalize probabilistic amplitude shaping (PAS) with binary codes [1] to the case of non-binary codes defined over prime finite fields. Firstly, we introduce probabilistic shaping via time sharing where shaping applies to information symbols only. Then, we design circular quadrature amplitude modulations (CQAM) that allow to directly generalize PAS to prime finite fields with full shaping.

Given a synchronous wireless network with N nodes uniformly located at random on a finite plane, we consider the problem of distributed peer discovery: all nodes want to discover as many other nodes as possible. We assume that there are a total of K physical resources dedicated for the discovery purpose. Each node can pick one resource to transmit its node identifier on, and can receive on the remaining K-1 resources. We assume that node identifiers are broadcast via coded transmission on one single physical resource. This paper addresses link level strategies to increase the average number of discovered devices when N ≅ aK, where a is the degree of density of the network. A classical strategy of discovering one node per resource would lead to a maximum of K-1 nodes being discovered. We focus on a multiple-access channel (MAC) scernario where multiple interferer users are jointly decoded. We propose a scheme to improve the performance. The method, based on iterative belief propagation on factor graph, is called joint iterative decoding (JID). It is shown, through system simulations, that JID may gain by significantly more than 100% over the classical single-user decoding, and by 20-25% over the successive interference cancellation (SIC).

There is a fundamental relationship between belief propagation (BP) and maximuma posterioridecoding. The case of transmission over the binary erasure channel was investigated in detail in a companion paper (C. MEacuteasson, A. Montanari, and R. Urbanke, "Maxwell's construction: The hidden bridge between iterative and maximum a posteriori decoding,"IEEE Transactions on Information Theory, submitted for publication). This paper investigates the extension to general memoryless channels (paying special attention to the binary case). An area theorem for transmission over general memoryless channels is introduced and some of its many consequences are discussed. We show that this area theorem gives rise to an upper bound on the maximuma posteriorithreshold for sparse graph codes. In situations where this bound is tight, the extrinsic soft bit estimates delivered by the BP decoder coincide with the correcta posterioriprobabilities above the maximuma posteriorithreshold. More generally, it is conjectured that the fundamental relationship between the maximuma posterioriprobability (MAP) and the BP decoder which was observed for transmission over the binary erasure channel carries over to the general case. We finally demonstrate that in order for the design rate of an ensemble to approach the capacity under BP decoding the component codes have to be perfectly matched, a statement which is well known for the special case of transmission over the binary erasure channel.

There is a fundamental relationship between belief propagation and maximuma posterioridecoding. A decoding algorithm, which is called the Maxwell decoder, is introduced and provides a constructive description of this relationship. Both the algorithm itself and the analysis of the new decoder are reminiscent of the Maxwell construction in thermodynamics. This paper investigates in detail the case of transmission over the binary erasure channel, while the extension to general binary memoryless channels is discussed in a companion paper.

The rate of a code is one of its most important parameters. We consider sparse graph codes and ask whether the rate of a random element of an ensemble is typically close to the design rate of the ensemble. For regular LDPC ensembles this question was answered in the affirmative in (Miller and Cohen, 2003). We start by giving an alternative proof of this statement. We then show that essentially the same type of argument applies not only to regular ensembles but also to ensembles that are derived from regular ensembles in the sense that their degree distribution is the result of applying the peeling decoder to a regular code. As an immediate consequence we prove that for regular ensembles the asymptotic MAP EXIT value coincides with the asymptotic BP EXIT value. We then give a systematic construction of ensembles for which rate and design rate differ. To accomplish this, we first show that the duality theorem (Ashikhminet al., 2004) implies that the asymptotic BP EXIT and the MAP EXIT functions are identical for any channel parameter for which the density evolution (DE) equations have a unique fixed point.

We derive further properties of EXIT and generalized EXIT curves. In particular we present an area theorem for iterative (as compared to MAP) decoding, we show how to compute upper-bounds on the MAP threshold for general channels and we apply these techniques to turbo codes

Consider transmission over the binary erasure channel using LDPC codes. We observed in [C. Measson, et al. (2003)] a curious relationship between the resulting maximum-likelihood (ML) decoding curve and the related performance curve under iterative (IT) decoding. We interpret this as an instance of Maxwell's construction which in its original form expresses an equilibrium condition at a phase transition.