Carlo Condo

A Fast Polar Code List Decoder Architecture Based on Sphere Decoding

Polar codes are a recently discovered family of capacity-achieving error-correcting codes. Among the proposed decoding algorithms, successive-cancellation list decoding guarantees the best error-correction performance with codes of moderate lengths, but it yields low throughput. Speed-up techniques have been proposed in the past: most of them rely on approximations that degrade the error-correction capability of the algorithm. We propose a speed-up technique for successive-cancellation list decoding of polar codes that is exact for list size of 2, while its approximations bring negligible error-correction performance degradation (<;0.05 dB) for other list sizes. A decoder architecture is designed: the proposed technique increases the throughput of a factor of 3.16×, at the cost of 14.2% in area occupation.

Simplified Successive-Cancellation List decoding of polar codes

The Successive-Cancellation List (SCL) decoding algorithm is one of the most promising approaches towards practical polar code decoding. It is able to provide a good trade-off between error-correction performance and complexity, tunable through the size of the list. In this paper, we show that in the conventional formulation of SCL, there are redundant calculations which do not need to be performed in the course of the algorithm. We simplify SCL by removing these redundant calculations and prove that the proposed simplified SCL and the conventional SCL algorithms are equivalent. The simplified SCL algorithm is valid for any code and can reduce the time-complexity of SCL without affecting the space complexity.

List sphere decoding of polar codes

Polar codes have gained a lot of attention during the past few years, because they can provably achieve the capacity of a memoryless channel. The design of efficient polar code decoders has been an active topic of research. The simple Successive Cancellation (SC) decoding algorithm yields poor error correction performance on short polar codes: the SC- List (SCL) algorithm overcomes this problem, but its hardware implementation requires a large amount of memory. Sphere Decoding (SD) is an alternative decoding technique that has been shown to work well for short polar codes, but it is burdened by undesirable characteristics. The performance of SD strongly depends on the choice of a suitable sphere radius, whose value must be selected according to the conditions of the channel. Channel conditions also affect the algorithms time complexity, that is consequently variable. In this paper, we introduce a List- SD algorithm for short polar codes. It has a fixed time complexity and does not make use of a radius: thus, no knowledge of the channel noise level is required. It is shown that the error correction performance of List-SD can match that of SC and SCL with as low as 72% of their memory requirements.

Sparse superposition codes: A practical approach

Sparse Superposition Codes are a class of capacity achieving codes for which decoding can be interpreted as a compressive sensing problem. The approximate message passing algorithm, proven to be effective in compressive sensing, has been proposed in different incarnations as a valid decoding approach. However, most literature focuses on infinite code length and asymptotic performance, while the strong reliance on matrix-and vector-wise operations suggests that a hardware-oriented approach might be more efficient. This work analyzes the performance of two decoding algorithms with finite code lengths and fixed point precision: 5-bit codeword symbol quantization is shown to cause performance degradation ≤ 0.15 dB. In-algorithm quantization values are proposed, together with code construction and algorithm approximations that cause negligible performance degradation. After selecting a set of codes as a case study, a decoding complexity estimation is performed, demonstrating that a fully parallel architecture is unfeasible. Suggestions and improvements towards partially-parallel solutions are given.

Blind Detection With Polar Codes

In blind detection, a set of candidates has to be decoded within a strict time constraint, to identify which transmissions are directed at the user equipment. We propose a blind detection scheme based on polar codes, where the radio network temporary identifier is transmitted instead of some of the frozen bits. A low-complexity decoding phase decodes all candidates, selecting a subset that is decoded by a high-performance algorithm. Simulations results show good missed detection and false alarm rates, that meet the 3GPP LTE-A and future 5G standard specifications. We also propose an early stopping criterion that can reduce the number of operations performed, improving both average latency and energy consumption. The detection speed is analyzed and different system parameter combinations are shown to meet the stringent timing requirements, leading to various implementation trade-offs.

Matrix reordering for efficient list sphere decoding of polar codes

The Successive-Cancellation List (SCL) algorithm is one of the best polar code decoding algorithms in terms of trade-offs between complexity and error correction performance. The List-Sphere Decoding (List-SD) algorithm has been recently proposed: it yields a better complexity/performance trade-off than SCL in the decoding of short polar codes, that can be used as component codes for larger polar codes. We exploit the structure of the generator matrix of polar codes to propose a matrix reordering technique which allows to significantly reduce the List-SD complexity without degrading its error correction performance, further improving the aforementioned trade-off. The proposed technique is implemented on hardware and it is shown that at the same Frame Error Rate (FER) and Bit Error Rate (BER), the matrix reordering can reduce the resource requirements of List-SD of up to 73%. Furthermore, FER and BER curves are plotted for case studies, showing that at the same complexity cost, matrix reordering improves the performance of List-SD of up to 0.75 dB at FER=10-2.

Fast Simplified Successive-Cancellation List Decoding of Polar Codes

Polar codes are capacity achieving error correcting codes that can be decoded through the successive-cancellation algorithm. To improve its error-correction performance, a list-based version called successive-cancellation list (SCL) has been proposed in the past, that however substantially increases the number of time-steps in the decoding process. The simplified SCL (SSCL) decoding algorithm exploits constituent codes within the polar code structure to greatly reduce the required number of time-steps without introducing any error-correction performance loss. In this paper, we propose a faster decoding approach to decode one of these constituent codes, the Rate-1 node. We use this Rate-1 node decoder to develop Fast-SSCL. We demonstrate that only a list-size-bound number of bits needs to be estimated in Rate-1 nodes and Fast-SSCL exactly matches the error-correction performance of SCL and SSCL. This technique can potentially greatly reduce the total number of time-steps needed for polar codes decoding: analysis on a set of case studies show that Fast-SSCL has a number of time- steps requirement that is up to 66.6% lower than SSCL and 88.1% lower than SCL.

Stall pattern avoidance in polynomial product codes

Product codes are a concatenated error-correction scheme that has been often considered for applications requiring very low bit-error rates, which demand that the error floor be decreased as much as possible. In this work, we consider product codes constructed from polynomial algebraic codes, and propose a novel low-complexity post-processing technique that is able to improve the error-correction performance by orders of magnitude. We provide lower bounds for the error rate achievable under post processing, and present simulation results indicating that these bounds are tight.

Design and Implementation of a Polar Codes Blind Detection Scheme

In blind detection, a set of candidates has to be decoded within a strict time constraint, to identify which transmissions are directed at the user equipment. Blind detection is required by the 3GPP LTE/LTE-A and fifth generation (5G) standards. With the selection of polar codes in 5G, the issue of blind detection of polar codes needs to be addressed. A polar code blind detection scheme has been recently proposed where the user ID is transmitted instead of some of the frozen bits. We propose an architecture to implement an improved version of such scheme. A first, coarse decoding phase helps selecting a subset of candidates that is decoded by a more powerful algorithm: an early stopping criterion is also introduced for the second decoding phase. The architecture relies on a tunable decoder that can be used for both phases. The architecture is synthesized and implementation results are reported for various system parameters. The reported area occupation and latency, obtained in 65-nm CMOS technology, are able to meet 5G requirements.

Reduced-memory high-throughput fast-SSC polar code decoder architecture

Polar codes have been selected for use within 5G networks, and are being considered for data and control channel for additional 5G scenarios, like the next generation ultra reliable low latency channel. As a result, efficient fast polar code decoder implementations are essential. In this work, we present a new fast simplified successive cancellation (Fast-SSC) decoder architecture. Our proposed solution is able to reduce the memory requirements and has an improved throughput with respect to state of the art Fast-SSC decoders. We achieve these objectives through a more efficient memory utilization than that of Fast-SSC, which also enables to execute multiple instructions in a single clock cycle. Our work shows that, compared to the state of the art, memory requirements are reduced by 22.2%; at the same time, a throughput improvement of 11.6% is achieved with (1024, 512) polar codes. Comparing equal throughputs, the memory requirements are reduced by up to 60.4%.