Marco Mondelli

How to achieve the capacity of asymmetric channels

We describe coding techniques that achieve the capacity of a discrete memoryless asymmetric channel. To do so, we discuss how recent advances in coding for symmetric channels yield more efficient solutions also for the asymmetric case. In more detail, we consider three basic approaches. The first one is Gallagers scheme that concatenates a linear code with a non-linear mapper, in order to bias the input distribution. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Further, we derive a scaling law between the gap to capacity, the cardinality of channel input and output alphabets, and the required size of the mapper. The second one is an integrated approach in which the coding scheme is used both for source coding, in order to create codewords with the capacity-achieving 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 approach is based on an idea due to Böcherer and Mathar and separates completely the two tasks of source coding and channel coding by “chaining” together several codewords. We prove that we can combine any suitable source code with any suitable channel code in order to provide optimal schemes for asymmetric channels. In particular, polar codes and spatially coupled codes fulfill the required conditions.

Achieving Marton’s Region for Broadcast Channels Using Polar Codes

This paper presents polar coding schemes for the two-user discrete memoryless broadcast channel (DM-BC) which achieve Marton’s region with both common and private messages. This is the best achievable rate region known to date, and it is tight for all classes of two-user DM-BCs whose capacity regions are known. To accomplish this task, we first construct polar codes for both the superposition as well as binning strategy. By combining these two schemes, we obtain Marton’s region with private messages only. Finally, we show how to handle the case of common information. The proposed coding schemes possess the usual advantages of polar codes, i.e., they have low encoding and decoding complexity and a superpolynomial decay rate of the error probability. We follow the lead of Goela, Abbe, and Gastpar, who recently introduced polar codes emulating the superposition and binning schemes. To align the polar indices, for both schemes, their solution involves some degradedness constraints that are assumed to hold between the auxiliary random variables and channel outputs. To remove these constraints, we consider the transmission of

Reed-Muller codes achieve capacity on erasure channels

k

Achieving Marton's region for broadcast channels using polar codes

blocks and employ a chaining construction that guarantees the proper alignment of the polarized indices. The techniques described in this paper are quite general, and they can be adopted to many other multiterminal scenarios whenever there polar indices need to be aligned.

Unified scaling of polar codes: Error exponent, scaling exponent, moderate deviations, and error floors

We introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding. Rather than relying on the precise structure of the codes, our method exploits code symmetry. In particular, the technique applies to any sequence of linear codes where the blocklengths are strictly increasing, the code rates converge, and the permutation group of each code is doubly transitive. In other words, we show that symmetry alone implies near-optimal performance. An important consequence of this result is that a sequence of Reed–Muller codes with increasing blocklength and converging rate achieves capacity. This possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1. Moreover, these results extend naturally to all affine-invariant codes and, thus, to extended primitive narrow-sense BCH codes. This also resolves, in the affirmative, the existence question for capacity-achieving sequences of binary cyclic codes. The primary tools used in the proof are the sharp threshold property for symmetric monotone Boolean functions and the area theorem for extrinsic information transfer functions.

Scaling exponent of list decoders with applications to polar codes

This paper presents polar coding schemes for the two-user discrete memoryless broadcast channel (DM-BC) which achieve Martons region with both common and private messages. This is the best achievable rate region known to date, and it is tight for all classes of two-user DM-BCs whose capacity regions are known. To accomplish this task, we first construct polar codes for both the superposition as well as binning strategy. By combining these two schemes, we obtain Martons region with private messages only. Finally, we show how to handle the case of common information. The proposed coding schemes possess the usual advantages of polar codes, i.e., they have low encoding and decoding complexity and a superpolynomial decay rate of the error probability. We follow the lead of Goela, Abbe, and Gastpar, who recently introduced polar codes emulating the superposition and binning schemes. To align the polar indices, for both schemes, their solution involves some degradedness constraints that are assumed to hold between the auxiliary random variables and channel outputs. To remove these constraints, we consider the transmission of k blocks and employ a chaining construction that guarantees the proper alignment of the polarized indices. The techniques described in this paper are quite general, and they can be adopted to many other multiterminal scenarios whenever there polar indices need to be aligned.

Construction of polar codes with sublinear complexity

Consider transmission of a polar code of block length N and rate R over a binary memoryless symmetric channel W with capacity I(W) and Bhattacharyya parameter Z(W) and let P_e be the error probability under successive cancellation decoding. Recall that in the error exponent regime, the channel W and R <; I(W) are fixed, while P_e scales roughly as 2^-√(N). In the scaling exponent regime, the channel W and P_e are fixed, while the gap to capacity I(W) - R scales as N^-1/μ, with 3.579 ≤ μ ≤ 5.702 for any W. We develop a unified framework to characterize the relationship between R, N, P_e, and W. First, we provide the tighter upper bound μ ≤ 4.714, valid for any W. Furthermore, when W is a binary erasure channel, we obtain an upper bound approaching very closely the value which was previously derived in a heuristic manner. Secondly, we consider a moderate deviations regime and we study how fast both the gap to capacity I(W) - R and the error probability P_e simultaneously go to 0 as N goes large. Thirdly, we prove that polar codes are not affected by error floors. To do so, we fix a polar code of block length N and rate R, we let the channel W vary, and we show that P_e scales roughly as Z(W)^√(N).

Joint Power Allocation and Path Selection for Multi-Hop Noncoherent Decode and Forward UWB Communications

Motivated by the significant performance gains which polar codes experience under successive cancellation list decoding, their scaling exponent is studied as a function of the list size. In particular, the error probability is fixed, and the tradeoff between the block length and back-off from capacity is analyzed. A lower bound is provided on the error probability under MAP decoding with list size L for any binary-input memoryless output-symmetric channel and for any class of linear codes such that their minimum distance is unbounded as the block length grows large. Then, it is shown that under MAP decoding, although the introduction of a list can significantly improve the involved constants, the scaling exponent itself, i.e., the speed at which capacity is approached, stays unaffected for any finite list size. In particular, this result applies to polar codes, since their minimum distance tends to infinity as the block length increases. A similar result is proved for genie-aided successive cancellation decoding when transmission takes place over the binary erasure channel, namely, the scaling exponent remains constant for any fixed number of helps from the genie. Note that since genie-aided successive cancellation decoding might be strictly worse than successive cancellation list decoding, the problem of establishing the scaling exponent of the latter remains open.

Reed–Muller Codes Achieve Capacity on Erasure Channels

Consider the problem of constructing a polar code of block length N for the transmission over a given channel W. Typically this requires to compute the reliability of all the N synthetic channels and then to include those that are sufficiently reliable. However, we know from [1], [2] that there is a partial order among the synthetic channels. Hence, it is natural to ask whether we can exploit it to reduce the computational burden of the construction problem. We show that, if we take advantage of the partial order [1], [2], we can construct a polar code by computing the reliability of roughly N/ log3/2 N synthetic channels. Such a set of synthetic channels is universal, in the sense that it allows one to construct polar codes for any W, and it can be identified by solving a maximum matching problem on a bipartite graph. Our proof technique consists in reducing the construction problem to the problem of computing the maximum cardinality of an antichain for a suitable partially ordered set. As such, this method is general and it can be used to further improve the complexity of the construction problem in case a new partial order on the synthetic channels of polar codes is discovered.

Comparing the bit-MAP and block-MAP decoding thresholds of reed-muller codes on BMS channels

With the aim of extending the coverage and improving the performance of impulse radio ultra-wideband (UWB) systems, this paper focuses on developing a novel single differential encoded decode and forward (DF) non-cooperative relaying scheme (NCR). To favor simple receiver structures, differential noncoherent detection is employed which enables effective energy capture without any channel estimation. Putting emphasis on the general case of multi-hop relaying, we illustrate an original algorithm for the joint power allocation and path selection (JPAPS), minimizing an approximate expression of the overall bit error rate (BER). In particular, after deriving a closed-form power allocation strategy, the optimal path selection is reduced to a shortest path problem on a connected graph, which can be solved without any topology information with complexity O(N3), N being the number of available relays of the network. An approximate scheme is also presented, which reduces the complexity to O(N2) while showing a negligible performance loss, and for benchmarking purposes, an exhaustive-search based multi-hop DF cooperative strategy is derived. Simulation results for various network setups corroborate the effectiveness of the proposed low-complexity JPAPS algorithm, which favorably compares to existing AF and DF relaying methods.