arXiv subjects

Ingmar Land

Publications and source records attributed to Ingmar Land.

At least 19 recordsLinked to original sources

arXiv2022

Group Properties of Polar Codes for Automorphism Ensemble Decoding

In this paper, we propose an analysis of the automorphism group of polar codes, with the scope of designing codes tailored for automorphism ensemble (AE) decoding. We prove the equivalence between the notion of decreasing monomial codes and the universal partial order (UPO) framework for the description of polar codes. Then, we analyze the algebraic properties of the affine automorphisms group of polar codes, providing a novel description of its structure and proposing a classification of automorphisms providing the same results under permutation decoding. Finally, we propose a method to list all the automorphisms that may lead to different candidates under AE decoding; by introducing the concept of redundant automorphisms, we find the maximum number of permutations providing possibly different codeword candidates under AE-SC, proposing a method to list all of them. A numerical analysis of the error correction performance of AE algorithm for the decoding of polar codes concludes the paper.

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arXiv2021

Tensor Decomposition Bounds for TBM-Based Massive Access

Tensor-based modulation (TBM) has been proposed in the context of unsourced random access for massive uplink communication. In this modulation, transmitters encode data as rank-1 tensors, with factors from a discrete vector constellation. This construction allows to split the multi-user receiver into a user separation step based on a low-rank tensor decomposition, and independent single-user demappers. In this paper, we analyze the limits of the tensor decomposition using Cram\'er-Rao bounds, providing bounds on the accuracy of the estimated factors. These bounds are shown by simulation to be tight at high SNR. We introduce an approximate perturbation model for the output of the tensor decomposition, which facilitates the computation of the log-likelihood ratios (LLR) of the transmitted bits, and provides an approximate achievable bound for the finite-length error probability. Combining TBM with classical forward error correction coding schemes such as polar codes, we use the approximate LLR to derive soft-decision decoder showing a gain over hard-decision decoders at low SNR.

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arXiv2021

Classification of Automorphisms for the Decoding of Polar Codes

This paper proposes new polar code design principles for the low-latency automorphism ensemble (AE) decoding. Our proposal permits to design a polar code with the desired automorphism group (if possible) while assuring the decreasing monomial property. Moreover, we prove that some automorphisms are redundant under AE decoding, and we propose a new automorphisms classification based on equivalence classes. Finally, we propose an automorphism selection heuristic based on drawing only one element of each class; we show that this method enhances the block error rate (BLER) performance of short polar codes even with a limited number of automorphisms.

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arXiv2021

High-performance low-complexity error pattern generation for ORBGRAND decoding

Guessing Random Additive Noise Decoding (GRAND) is a recently proposed decoding method searching for the error pattern applied to the transmitted codeword. Ordered reliability bit GRAND (ORBGRAND) uses soft channel information to reorder entries of error patterns, generating them according to a fixed schedule, i.e. their logistic weight. In this paper, we show that every good ORBGRAND scheduling should follow an universal partial order, and we present an algorithm to generate the logistic weight order accordingly. We then propose an improved error pattern schedule that can improve the performance of ORBGRAND of 0.5dB at a block error rate (BLER) of $10^{-5}$, with increasing gains as the BLER decreases. This schedule can be closely approximated with a low-complexity generation algorithm that is shown to incur no BLER degradation.

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arXiv2021

Staircase codes with non-systematic polar codes

In this work we propose an encoding and decoding framework for staircase codes based on non-systematic polar codes as component codes. The staircase structure allows for efficient parallelized decoding, while the polar component codes allow to benefit from the flexible structure and efficient soft-decision decoding algorithms. To enhance the performance of the polar staircase codes, we concatenate the polar component codes with cyclic redundancy check (CRC) outer codes, and we add interleavers within the staircase structure that are specific to polar code properties. The CRCs also allow to substantially reduce the decoding complexity. Simulation results evaluate the gain brought by our proposed techniques, and analyze the dependence of the error-correction performance on code and decoder parameters. Comparison with the state of the art on staircase polar codes shows an improvement in BER up to 0.9~dB, or considerable complexity reduction at the same BER.

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arXiv2021

Polar Codes for Automorphism Ensemble Decoding

In this paper we deal with polar code automorphisms that are beneficial under low-latency automorphism ensemble (AE) decoding, and we propose polar code designs that have such automorphisms. Successive-cancellation (SC) decoding and thus SC-based AE decoding are invariant with respect to the only known polar code automorphisms, namely those of the lower-triangular affine (LTA) group. To overcome this problem, we provide methods to determine whether a given polar code has non-LTA automorphisms and to identify such automorphisms. Building on this, we design specific polar codes that admit automorphisms in the upper-diagonal linear (UTL) group, and thus render SC-based AE decoding effective. Demonstrated by examples, these new polar codes under AE decoding outperform conventional polar codes under SC list decoding in terms of error rate, while keeping the latency comparable to SC decoding. Moreover, state-of-the-art BP-based permutation decoding for polar codes is beaten by BP-based AE thanks to this design.

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arXiv2020

Tensor-Based Modulation for Unsourced Massive Random Access

We introduce a modulation for unsourced massive random access whereby the transmitted symbols are rank-1 tensors constructed from Grassmannian sub-constellations. The use of a low-rank tensor structure, together with tensor decomposition in order to separate the users at the receiver, allows a convenient uncoupling between multi-user separation and single-user demapping. The proposed signaling scheme is designed for the block fading channel and multiple-antenna settings, and is shown to perform well in comparison to state-of-the-art unsourced approaches.

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arXiv2020

Sliding Window Polar Codes

We propose a novel coupling technique for the design of polar codes of length N, making them decodable through a sliding window of size M < N. This feature allows to reduce the computational complexity of the decoder, an important possibility in wireless communication downlink scenarios. Our approach is based on the design of an ad-hoc kernel to be inserted in a multi-kernel polar code framework; this structure enables the sliding window decoding of the code. Simulation results show that the proposed sliding window polar codes outperform the independent blocks transmission in the proposed scenario, at the cost of a negligible decoding overhead.

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arXiv2020

Multi-Kernel Polar Codes: Concept and Design Principles

In this paper, we propose a new polar code construction by employing kernels of different sizes in the Kronecker product of the transformation matrix, thus generalizing the original construction by Arikan. The proposed multi-kernel polar code allows for more flexibility in terms of the code length, moreover allowing for various new design principles. We describe in detail encoding as well as successive cancellation (SC) decoding and SC list (SCL) decoding, and we provide a novel design method for the frozen set that allows to optimise the performance under list decoding, as opposed to original relability-based code design. Finally, we numerically demonstrate the advantage of multi-kernel polar codes under the new design principles compared to punctured and shortened polar codes.

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arXiv2019

Practical Product Code Construction of Polar Codes

In this paper, we study the connection between polar codes and product codes. Our analysis shows that the product of two polar codes is again a polar code, and we provide guidelines to compute its frozen set on the basis of the frozen sets of the component polar codes. Moreover, we show how polar codes can be described as irregular product codes. We propose a two-step decoder for long polar codes taking advantage of this dual nature to heavily reduce decoding latency. Finally, we show that the proposed decoding technique outperforms both standard polar codes and state-of-the-art codes for optical communications under latency constraints.

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arXiv2019

Partially Ordered Statistics Demapping for Multi-Dimensional Modulation Formats

We propose a very low-complexity and high-performance algorithm for soft-demapping of multi-dimensional modulation formats. We assess its performance over the linear channel for four 8D formats, generated using binary arithmetics. This solution outperforms current algorithms in terms of complexity without loss in performances.

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arXiv2019

Construction and Decoding of Product Codes with Non-Systematic Polar Codes

Product codes are widespread in optical communications, thanks to their high throughput and good error-correction performance. Systematic polar codes have been recently considered as component codes for product codes. In this paper, we present a novel construction for product polar codes based on non-systematic polar codes. We prove that the resulting product code is actually a polar code, having a frozen set that is dependent on the frozen sets of the component polar codes. We propose a low-complexity decoding algorithm exploiting the dual nature of the constructed code. Performance analysis and simulations show high decoding speed, that allows to construct long codes while maintaining low decoding latency. The resulting high throughput and good error-correction performance are appealing for optical communication systems and other systems where high throughput and low latency are required.

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arXiv2018

Memory Management in Successive-Cancellation based Decoders for Multi-Kernel Polar Codes

Multi-kernel polar codes have recently been proposed to construct polar codes of lengths different from powers of two. Decoder implementations for multi-kernel polar codes need to account for this feature, that becomes critical in memory management. We propose an efficient, generalized memory management framework for implementation of successivecancellation decoding of multi-kernel polar codes. It can be used on many types of hardware architectures and different flavors of SC decoding algorithms. We illustrate the proposed solution for small kernel sizes, and give complexity estimates for various kernel combinations and code lengths.

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arXiv2018

Generalized Fast Decoding of Polar Codes

Research on polar codes has been constantly gaining attention over the last decade, by academia and industry alike, thanks to their capacity-achieving error-correction performance and low-complexity decoding algorithms. Recently, they have been selected as one of the coding schemes in the $5^{th}$ generation wireless standard (5G). Over the years various polar code decoding algorithms, like SC-list (SCL), have been proposed to improve the mediocre performance of the successive cancellation (SC) decoding algorithm for finite code lengths; however, like SC, they suffer from long decoding latency. Fast decoding of polar codes tries to overcome this problem by identifying particular subcodes in the polar code and decoding them with efficient decoders. In this work, we introduce a generalized approach to fast decoding of polar codes to further reduce SC-based decoding latency. We propose three multi-node polar code subcodes whose identification patterns include most of the existing subcodes, extending them to SCL decoding, and allow to apply fast decoding to larger subsets of bits. Without any error-correction performance degradation, the proposed technique shows up to $23.6\%$ and $29.2\%$ decoding latency gain with respect to fast SC and SCL decoding algorithms, respectively, and up to $63.6\%$ and $49.8\%$ if a performance loss is accepted, whose amount depends on code and decoding algorithm parameters, along with the desired speedup.

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arXiv2018

Design of Polar Codes in 5G New Radio

Polar codes have attracted the attention of academia and industry alike in the past decade, such that the 5$^\text{th}$ generation wireless systems (5G) standardization process of the 3$^\text{th}$ generation partnership project (3GPP) chose polar codes as a channel coding scheme. In this tutorial, we provide a description of the encoding process of polar codes adopted by the 5G standard. We illustrate the struggles of designing a family of polar codes able to satisfy the demands of 5G systems, with particular attention to rate flexibility and low decoding latency. The result of these efforts is an elaborate framework that applies novel coding techniques to provide a solid channel code for NR requirements.

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arXiv2017

Multi-Kernel Polar Codes: Proof of Polarization and Error Exponents

In this paper, we investigate a novel family of polar codes based on multi-kernel constructions, proving that this construction actually polarizes. To this end, we derive a new and more general proof of polarization, which gives sufficient conditions for kernels to polarize. Finally, we derive the convergence rate of the multi-kernel construction and relate it to the convergence rate of each of the constituent kernels.

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arXiv2017

$\beta$-expansion: A Theoretical Framework for Fast and Recursive Construction of Polar Codes

In this work, we introduce $\beta$-expansion, a notion borrowed from number theory, as a theoretical framework to study fast construction of polar codes based on a recursive structure of universal partial order (UPO) and polarization weight (PW) algorithm. We show that polar codes can be recursively constructed from UPO by continuously solving several polynomial equations at each recursive step. From these polynomial equations, we can extract an interval for $\beta$, such that ranking the synthetic channels through a closed-form $\beta$-expansion preserves the property of nested frozen sets, which is a desired feature for low-complex construction. In an example of AWGN channels, we show that this interval for $\beta$ converges to a constant close to $1.1892 \approx 2^{1/4}$ when the code block-length trends to infinity. Both asymptotic analysis and simulation results validate our theoretical claims.

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arXiv2017

Minimum-Distance Based Construction of Multi-Kernel Polar Codes

In this paper, we propose a construction for multi-kernel polar codes based on the maximization of the minimum distance. Compared to the original construction based on density evolution, our new design shows particular advantages for short code lengths, where the polarization effect has less impact on the performance than the distances of the code. We introduce and compute the minimum-distance profile and provide a simple greedy algorithm for the code design. Compared to state-of-the-art punctured or shortened Arikan polar codes, multi-kernel polar codes with our new design show significantly improved error-rate performance.

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