Construction of Polar Codes with Sublinear Complexity

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 a fraction $1/\log^{3/2} N$ of the synthetic channels. In particular, we prove that $N/\log^{3/2} N$ is a lower bound on the number of synthetic channels to be considered and such a bound is tight up to a multiplicative factor $\log\log N$. This set of roughly $N/\log^{3/2} N$ 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 of 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.Comment: 9 pages, 3 figures, presented at ISIT'17 and submitted to IEEE Trans. Inform. Theor

Partitioned List Decoding of Polar Codes: Analysis and Improvement of Finite Length Performance

Polar codes represent one of the major recent breakthroughs in coding theory and, because of their attractive features, they have been selected for the incoming 5G standard. As such, a lot of attention has been devoted to the development of decoding algorithms with good error performance and efficient hardware implementation. One of the leading candidates in this regard is represented by successive-cancellation list (SCL) decoding. However, its hardware implementation requires a large amount of memory. Recently, a partitioned SCL (PSCL) decoder has been proposed to significantly reduce the memory consumption. In this paper, we examine the paradigm of PSCL decoding from both theoretical and practical standpoints: (i) by changing the construction of the code, we are able to improve the performance at no additional computational, latency or memory cost, (ii) we present an optimal scheme to allocate cyclic redundancy checks (CRCs), and (iii) we provide an upper bound on the list size that allows MAP performance.Comment: 2017 IEEE Global Communications Conference (GLOBECOM

Sublinear Latency for Simplified Successive Cancellation Decoding of Polar Codes

This work analyzes the latency of the simplified successive cancellation (SSC) decoding scheme for polar codes proposed by Alamdar-Yazdi and Kschischang. It is shown that, unlike conventional successive cancellation decoding, where latency is linear in the block length, the latency of SSC decoding is sublinear. More specifically, the latency of SSC decoding is $O(N^{1-1/\mu})$, where $N$ is the block length and $\mu$ is the scaling exponent of the channel, which captures the speed of convergence of the rate to capacity. Numerical results demonstrate the tightness of the bound and show that most of the latency reduction arises from the parallel decoding of subcodes of rate $0$ or $1$.Comment: 20 pages, 6 figures, presented in part at ISIT 2020 and accepted in IEEE Transactions on Wireless Communication

How to Achieve the Capacity of Asymmetric Channels

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 B\"ocherer 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.Comment: 32 pages, 4 figures, presented in part at Allerton'14 and published in IEEE Trans. Inform. Theor

Rate-Flexible Fast Polar Decoders

Polar codes have gained extensive attention during the past few years and recently they have been selected for the next generation of wireless communications standards (5G). Successive-cancellation-based (SC-based) decoders, such as SC list (SCL) and SC flip (SCF), provide a reasonable error performance for polar codes at the cost of low decoding speed. Fast SC-based decoders, such as Fast-SSC, Fast-SSCL, and Fast-SSCF, identify the special constituent codes in a polar code graph off-line, produce a list of operations, store the list in memory, and feed the list to the decoder to decode the constituent codes in order efficiently, thus increasing the decoding speed. However, the list of operations is dependent on the code rate and as the rate changes, a new list is produced, making fast SC-based decoders not rate-flexible. In this paper, we propose a completely rate-flexible fast SC-based decoder by creating the list of operations directly in hardware, with low implementation complexity. We further propose a hardware architecture implementing the proposed method and show that the area occupation of the rate-flexible fast SC-based decoder in this paper is only $38\%$ of the total area of the memory-based base-line decoder when 5G code rates are supported

Parallelism versus Latency in Simplified Successive-Cancellation Decoding of Polar Codes

This paper characterizes the latency of the simplified successive-cancellation (SSC) decoding scheme for polar codes under hardware resource constraints. In particular, when the number of processing elements $P$ that can perform SSC decoding operations in parallel is limited, as is the case in practice, the latency of SSC decoding is $O\left(N^{1-1/\mu}+\frac{N}{P}\log_2\log_2\frac{N}{P}\right)$, where $N$ is the block length of the code and $\mu$ is the scaling exponent of the channel. Three direct consequences of this bound are presented. First, in a fully-parallel implementation where $P=\frac{N}{2}$, the latency of SSC decoding is $O\left(N^{1-1/\mu}\right)$, which is sublinear in the block length. This recovers a result from our earlier work. Second, in a fully-serial implementation where $P=1$, the latency of SSC decoding scales as $O\left(N\log_2\log_2 N\right)$. The multiplicative constant is also calculated: we show that the latency of SSC decoding when $P=1$ is given by $\left(2+o(1)\right) N\log_2\log_2 N$. Third, in a semi-parallel implementation, the smallest $P$ that gives the same latency as that of the fully-parallel implementation is $P=N^{1/\mu}$. The tightness of our bound on SSC decoding latency and the applicability of the foregoing results is validated through extensive simulations