List Decoding of Direct Sum Codes

We consider families of codes obtained by "lifting" a base code $\mathcal{C}$ through operations such as $k$-XOR applied to "local views" of codewords of $\mathcal{C}$, according to a suitable $k$-uniform hypergraph. The $k$-XOR operation yields the direct sum encoding used in works of [Ta-Shma, STOC 2017] and [Dinur and Kaufman, FOCS 2017]. We give a general framework for list decoding such lifted codes, as long as the base code admits a unique decoding algorithm, and the hypergraph used for lifting satisfies certain expansion properties. We show that these properties are satisfied by the collection of length $k$ walks on an expander graph, and by hypergraphs corresponding to high-dimensional expanders. Instantiating our framework, we obtain list decoding algorithms for direct sum liftings on the above hypergraph families. Using known connections between direct sum and direct product, we also recover the recent results of Dinur et al. [SODA 2019] on list decoding for direct product liftings. Our framework relies on relaxations given by the Sum-of-Squares (SOS) SDP hierarchy for solving various constraint satisfaction problems (CSPs). We view the problem of recovering the closest codeword to a given word, as finding the optimal solution of a CSP. Constraints in the instance correspond to edges of the lifting hypergraph, and the solutions are restricted to lie in the base code $\mathcal{C}$. We show that recent algorithms for (approximately) solving CSPs on certain expanding hypergraphs also yield a decoding algorithm for such lifted codes. We extend the framework to list decoding, by requiring the SOS solution to minimize a convex proxy for negative entropy. We show that this ensures a covering property for the SOS solution, and the "condition and round" approach used in several SOS algorithms can then be used to recover the required list of codewords.Comment: Full version of paper from SODA 202

Fixed Error Asymptotics For Erasure and List Decoding

We derive the optimum second-order coding rates, known as second-order capacities, for erasure and list decoding. For erasure decoding for discrete memoryless channels, we show that second-order capacity is $\sqrt{V}\Phi^{-1}(\epsilon_t)$ where $V$ is the channel dispersion and $\epsilon_t$ is the total error probability, i.e., the sum of the erasure and undetected errors. We show numerically that the expected rate at finite blocklength for erasures decoding can exceed the finite blocklength channel coding rate. We also show that the analogous result also holds for lossless source coding with decoder side information, i.e., Slepian-Wolf coding. For list decoding, we consider list codes of deterministic size that scales as $\exp(\sqrt{n}l)$ and show that the second-order capacity is $l+\sqrt{V}\Phi^{-1}(\epsilon)$ where $\epsilon$ is the permissible error probability. We also consider lists of polynomial size $n^\alpha$ and derive bounds on the third-order coding rate in terms of the order of the polynomial $\alpha$. These bounds are tight for symmetric and singular channels. The direct parts of the coding theorems leverage on the simple threshold decoder and converses are proved using variants of the hypothesis testing converse.Comment: 18 pages, 1 figure; Submitted to IEEE Transactions on Information Theory; Shorter version to be presented at ISIT 201

Homomorphism Extension

We define the Homomorphism Extension (HomExt) problem: given a group $G$, a subgroup $M \leq G$ and a homomorphism $\varphi: M \to H$, decide whether or not there exists a homomorphism $\widetilde{\varphi}: G\to H$ extending $\varphi$, i.e., $\widetilde{\varphi}|_M = \varphi$. This problem arose in the context of list-decoding homomorphism codes but is also of independent interest, both as a problem in computational group theory and as a new and natural problem in NP of unsettled complexity status. We consider the case $H=S_m$ (the symmetric group of degree $m$), i.e., $\varphi : G \to H$ is a $G$-action on a set of $m$ elements. We assume $G\le S_n$ is given as a permutation group by a list of generators. We characterize the equivalence classes of extensions in terms of a multidimensional oracle subset-sum problem. From this we infer that for bounded $G$ the HomExt problem can be solved in polynomial time. Our main result concerns the case $G=A_n$ (the alternating group of degree $n$) for variable $n$ under the assumption that the index of $M$ in $G$ is bounded by poly$(n)$. We solve this case in polynomial time for all $m < 2^{n-1}/\sqrt{n}$. This is the case with direct relevance to homomorphism codes (Babai, Black, and Wuu, arXiv 2018); it is used as a component of one of the main algorithms in that paper.Comment: 29 page

SISO APP Searches in Lattices with Tanner Graphs

An efficient, low-complexity, soft-output detector for general lattices is presented, based on their Tanner graph (TG) representations. Closest-point searches in lattices can be performed as non-binary belief propagation on associated TGs; soft-information output is naturally generated in the process; the algorithm requires no backtrack (cf. classic sphere decoding), and extracts extrinsic information. A lattice's coding gain enables equivalence relations between lattice points, which can be thereby partitioned in cosets. Total and extrinsic a posteriori probabilities at the detector's output further enable the use of soft detection information in iterative schemes. The algorithm is illustrated via two scenarios that transmit a 32-point, uncoded super-orthogonal (SO) constellation for multiple-input multiple-output (MIMO) channels, carved from an 8-dimensional non-orthogonal lattice (a direct sum of two 4-dimensional checkerboard lattice): it achieves maximum likelihood performance in quasistatic fading; and, performs close to interference-free transmission, and identically to list sphere decoding, in independent fading with coordinate interleaving and iterative equalization and detection. Latter scenario outperforms former despite the absence of forward error correction coding---because the inherent lattice coding gain allows for the refining of extrinsic information. The lattice constellation is the same as the one employed in the SO space-time trellis codes first introduced for 2-by-2 MIMO by Ionescu et al., then independently by Jafarkhani and Seshadri. Complexity is log-linear in lattice dimensionality, vs. cubic in sphere decoders.Comment: 15 pages, 6 figures, 2 tables, uses IEEEtran.cl

A hybrid partial sum computation unit architecture for list decoders of polar codes

Although the successive cancelation (SC) algorithm works well for very long polar codes, its error performance for shorter polar codes is much worse. Several SC based list decoding algorithms have been proposed to improve the error performances of both long and short polar codes. A significant step of SC based list decoding algorithms is the updating of partial sums for all decoding paths. In this paper, we first proposed a lazy copy partial sum computation algorithm for SC based list decoding algorithms. Instead of copying partial sums directly, our lazy copy algorithm copies indices of partial sums. Based on our lazy copy algorithm, we propose a hybrid partial sum computation unit architecture, which employs both registers and memories so that the overall area efficiency is improved. Compared with a recent partial sum computation unit for list decoders, when the list size $L=4$, our partial sum computation unit achieves an area saving of 23\% and 63\% for block length $2^{13}$ and $2^{15}$, respectively.Comment: 5 pages, presented at the 2015 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP

Symbol-Decision Successive Cancellation List Decoder for Polar Codes

Polar codes are of great interests because they provably achieve the capacity of both discrete and continuous memoryless channels while having an explicit construction. Most existing decoding algorithms of polar codes are based on bit-wise hard or soft decisions. In this paper, we propose symbol-decision successive cancellation (SC) and successive cancellation list (SCL) decoders for polar codes, which use symbol-wise hard or soft decisions for higher throughput or better error performance. First, we propose to use a recursive channel combination to calculate symbol-wise channel transition probabilities, which lead to symbol decisions. Our proposed recursive channel combination also has a lower complexity than simply combining bit-wise channel transition probabilities. The similarity between our proposed method and Arikan's channel transformations also helps to share hardware resources between calculating bit- and symbol-wise channel transition probabilities. Second, a two-stage list pruning network is proposed to provide a trade-off between the error performance and the complexity of the symbol-decision SCL decoder. Third, since memory is a significant part of SCL decoders, we propose a pre-computation memory-saving technique to reduce memory requirement of an SCL decoder. Finally, to evaluate the throughput advantage of our symbol-decision decoders, we design an architecture based on a semi-parallel successive cancellation list decoder. In this architecture, different symbol sizes, sorting implementations, and message scheduling schemes are considered. Our synthesis results show that in terms of area efficiency, our symbol-decision SCL decoders outperform both bit- and symbol-decision SCL decoders.Comment: 13 pages, 17 figure

Recursive Decoding and Its Performance for Low-Rate Reed-Muller Codes

Recursive decoding techniques are considered for Reed-Muller (RM) codes of growing length $n$ and fixed order $r.$ An algorithm is designed that has complexity of order $n\log n$ and corrects most error patterns of weight up to $n(1/2-\varepsilon)$ given that $\varepsilon$ exceeds $n^{-1/2^{r}}.$ This improves the asymptotic bounds known for decoding RM codes with nonexponential complexity

Fast Maximum-Likelihood Decoding of the Golden Code

The golden code is a full-rate full-diversity space-time code for two transmit antennas that has a maximal coding gain. Because each codeword conveys four information symbols from an M-ary quadrature-amplitude modulation alphabet, the complexity of an exhaustive search decoder is proportional to M^2. In this paper we present a new fast algorithm for maximum-likelihood decoding of the golden code that has a worst-case complexity of only O(2M^2.5). We also present an efficient implementation of the fast decoder that exhibits a low average complexity. Finally, in contrast to the overlaid Alamouti codes, which lose their fast decodability property on time-varying channels, we show that the golden code is fast decodable on both quasistatic and rapid time-varying channels.Comment: Submitted to IEEE Trans. on Wireless, November 200

Joint error correction enhancement of the fountain codes concept

Fountain codes like LT or Raptor codes, also known as rateless erasure codes, allow to encode a message as some number of packets, such that any large enough subset of these packets is sufficient to fully reconstruct the message. It requires undamaged packets, while the packets which were not lost are usually damaged in real scenarios. Hence, an additional error correction layer is often required: adding some level of redundancy to each packet to be able to repair eventual damages. This approach requires a priori knowledge of the final damage level of every packet - insufficient redundancy leads to packet loss, overprotection means suboptimal channel rate. However, the sender may have inaccurate or even no a priori information about the final damage levels, for example in applications like broadcasting, degradation of a storage medium or damage of picture watermarking. Joint Reconstruction Codes (JRC) setting is introduced and discussed in this paper for the purpose of removing the need of a priori knowledge of damage level and sub-optimality caused by overprotection and discarding underprotected packets. It is obtained by combining both processes: reconstruction from multiple packets and forward error correction. The decoder combines the resultant informational content of all received packets accordingly to their actual noise level, which can be estimated a posteriori individually for each packet. Assuming binary symmetric channel (BSC) of $\epsilon$ bit-flip probability, every potentially damaged bit carries $R_0(\epsilon)=1-h_1(\epsilon)$ bits of information, where $h_1$ is the Shannon entropy. The minimal requirement to fully reconstruct the message is that the sum of rate $R_0(\epsilon)$ over all bits is at least the size of the message. We will discuss sequential decoding for the reconstruction purpose, which statistical behavior can be estimated using Renyi entropy.Comment: 14 pages, 9 figure

Reduce the Complexity of List Decoding of Polar Codes by Tree-Pruning

Polar codes under cyclic redundancy check aided successive cancellation list (CA-SCL) decoding can outperform the turbo codes and the LDPC codes when code lengths are configured to be several kilobits. In order to reduce the decoding complexity, a novel tree-pruning scheme for the \mbox{SCL/CA-SCL} decoding algorithms is proposed in this paper. In each step of the decoding procedure, the candidate paths with metrics less than a threshold are dropped directly to avoid the unnecessary computations for the path searching on the descendant branches of them. Given a candidate path, an upper bound of the path metric of its descendants is proposed to determined whether the pruning of this candidate path would affect frame error rate (FER) performance. By utilizing this upper bounding technique and introducing a dynamic threshold, the proposed scheme deletes the redundant candidate paths as many as possible while keeping the performance deterioration in a tolerant region, thus it is much more efficient than the existing pruning scheme. With only a negligible loss of FER performance, the computational complexity of the proposed pruned decoding scheme is only about $40\%$ of the standard algorithm in the low signal-to-noise ratio (SNR) region (where the FER under CA-SCL decoding is about $0.1 \sim 0.001$), and it can be very close to that of the successive cancellation (SC) decoder in the moderate and high SNR regions