This paper proposes a syndrome sphere decoding (SSD) algorithm. SSD achieves the frame error rate (FER) of maximum likelihood (ML) decoding for a tail-biting convolutional code (TBCC) concatenated with an expurgating linear function (ELF) with significantly lower average and maximum decoding complexity than serial list Viterbi decoding (SLVD). SSD begins by using a Viterbi decoder to find the closest trellis path, which may not be tail-biting and may not satisfy the ELF. This trellis path has a syndrome comprised of the difference between the received ELF and the ELF computed from the received message and the difference between the beginning and ending state of the trellis path. This syndrome is used to find all valid tail-biting codewords that satisfy the ELF constraint and lie within a specified distance from the closest trellis path. The proposed algorithm avoids the complexity of SLVD at the cost of a large table containing all the offsets needed for each syndrome. A hybrid decoder combines SSD and SLVD with a fixed maximum list size, balancing the maximum list size for SLVD against the size of the SSD offset table.
Nanopore sequencers are a strong contender for the practical retrieval and decoding of information stored in DNA. In this paper, we provide two graph constructions of codes that satisfy the restrictions on sequences desirable for nanopore sequencers including GC-content, run length limit, and secondary structure avoidance. Moreover, we show that our constructions have error detecting capability with respect to single deletions, insertions, and substitutions.
We show that optimized message update scheduling can enhance both the error-rate performance and the throughput of layered BP decoding of LDPC codes. As an exemplary attempt, optimized scheduling accelerates BP decoding and leads to approximately 2x throughput enhancement when compared to top-to-bottom row-layered BP decoding of 5G LDPC codes. Furthermore, scheduling enhances the error-rate performance of short-length LDPC codes via ensemble decoding. For short-length 5G LDPC codes, coding gains of up to 0.7 dB and 1.6 dB at BLER of 10(-3)-10(-4) for both AWGN and Rayleigh fading channels can be achieved when compared to conventional BP decoders. Additionally, we show that the same concept applies to general linear block codes (e.g., polar/RM codes) facilitating a unified decoding architecture based on row-layered BP decoders ("one-silicon fits all").
This article studies the Hermitian hulls of linear codes from algebraic plane curves of a special type. We provide lower bounds on the hull dimension of one-point algebraic geometry codes. We derive two families of entanglement-assisted quantum codes from one-point algebraic geometry codes, using the Hermitian approach.
We propose a generalization of the recently proposed quantum Tanner codes by Leverrier and Zemor. These codes can be constructed equivalently from groups, Cayley graphs, or square complexes constructed from groups. In a recent work, we enlarged this to group actions on finite sets, Schreier graphs, and a family of square complexes. We extend the class of quantum Tanner codes further by replacing the tensor product code in the construction with a Tanner code on any bipartite graph. A stricter property on the other underlying graphs is required, and we show that a common variation of the construction can always be taken to satisfy this condition. This results in improved codes compared to the ones constructed based on Schreier graphs.
In this paper, we prove that the binary images of generalized Reed-Solomon (RS) codes are capacity-achieving over binary-input output-symmetric channels, in terms of frame error rate (FER) under maximum likelihood (ML) decoding. In the finite-length region, we propose to decode the generalized RS codes using the ordered statistics decoding with local constraints (LC-OSD) algorithm. Unlike LC-OSD of other codes, we replace Gaussian elimination with the parallel Lagrange interpolation plus change-of-basis to reduce the decoding latency. Simulation results show that the generalized RS codes can approach the random coding union bounds in a wide range of code rates.
Automorphism ensemble decoding (AED) with the successive cancellation (SC) constituent decoder (AED-SC) can achieve similar decoding performance to the successive cancellation list (SCL) decoder when decoding short-to-medium-length polar codes. However, implementing automorphisms requires additional hardware for routing, leading to significant area overhead, especially with a large number of automorphisms. We propose methods for selecting automorphisms in SC decoders to reduce routing overhead. We establish the equivalence between automorphism selection and the NP-hard minimum K-union (MKU) problem. To maintain decoding performance under the selected automorphisms, we formulate the equivalence class property of AED-SC as a quadratic constraint. Compared to the state-of-the-art, our method achieves up to a 2.9x reduction in the number of routes needed to implement all automorphisms.
Ordered statistics decoding (OSD) is a universal decoding algorithm for short-length linear block codes that approaches maximum likelihood decoding when run at sufficiently high order. However, the high computational complexity and latency of Gaussian elimination make it challenging for OSD implementations to meet stringent latency constraints. In this paper, we design a hardware decoder for OSD with order one, featuring an unfolded Gaussian elimination architecture with configurable parallelism. The proposed architecture can efficiently process multiple matrix columns and guarantees a low decoding latency. Based on 65 nm CMOS technology, the synthesis results show a worst-case throughput of 958 Mbps for a (128, 105) polar code at a frequency of 575 MHz.
This work investigates channel code design for next-generation data center networks employing 448G/lane attachment unit interfaces (AUIs). A three-link problem, composed of an electrical, optical, and electrical link, representative of intra- and inter-data center links ranging from a few meters to a few kilometers, is considered. To address high error rates, serially concatenated Reed Solomon (RS) and Bose-Ray-Chaudhuri-Hocquenghem (BCH) codes, both located on the host chip, are proposed. The implementation complexity of different forward error correction (FEC) component codes is discussed, different decoding schemes are compared, and analytical and semi-analytical approximations of the error rates are presented.
Due to code degeneracy and the graph structure of quantum low-density parity-check (QLDPC) codes, the performance of conventional belief propagation (BP) decoding can be poor. Recently, belief propagation guided decimation (BPGD) has shown promise to combat these challenges. In this paper, we investigate various decimation approaches to improve the error correcting performance and convergence speed of BPGD. We first consider soft decimation, where the BP equations are modified via several tuneable parameters. This approach exhibits linear complexity relative to the length of the block code and is shown to outperform hard decimation approaches for careful selection of the algorithm parameters. We then combine the approaches in a "soft-hard" BPGD variant, where hard decisions are periodically made and those symbols are permanently fixed throughout the remainder of the decoding process. Simulation results show that further performance improvement can be observed in this case at the cost of increasing the algorithmic complexity.
This paper proposes and studies a scheme of coded secure delivery (CSD), where data objects are delivered upon request in a way that only users that are allowed access can recover them. The security of the scheme is driven by encryption with keys that are pre-distributed to different subsets of the users. As a result, the system can securely serve anonymous requests, enhancing its users’ privacy. A CSD instance is defined formally by two binary matrices, and our main result is an algorithm to minimize the number of transmissions needed to accomplish it. Coding allows transmission of linear combinations of objects, and the power of coding is demonstrated by showing instance families with large gaps to uncoded delivery. In addition to minimizing communication costs, the paper studies the key-distribution problem toward minimizing the users’ storage costs.
Polar codes with large kernels offer improved error-correction performance but suffer from high decoding complexity due to costly marginalization operations. In this work, we propose neural network (NN)-based approximations of these marginalization functions, enabling efficient decoding for arbitrary linear kernels. We show that training on all-zero partial codewords is sufficient for linear kernels, significantly reducing the dataset size and allowing for smaller, faster NN models. Our method is evaluated within a Successive Cancellation (SC) decoder, being compatible with any blocklength, code rate, or SNR. Experimental results for kernel size 16 demonstrate strong performance and substantial complexity savings over exact marginalization.
We present a trainable and adaptive detector (the T-Detector) with a core trainable BCJR (T-BCJR) that leverages the extrinsic information provided by an outer SISO decoder to enhance its tracking capability in time-varying channels. The T-Detector is designed to be channel- and modulation-agnostic, performing key detection tasks such as channel shortening, Maximum Likelihood (ML) sequence detection, and Log-Likelihood (LL) computation for soft-input channel decoding. Its processing complexity remains equal to that of conventional detection methods. The exploitation of the outer decoder enhances system performance and reduces pilot overhead.
This paper introduces a convolutional syndrome former (CSF) which enables reduced complexity decoding of regularly punctured convolutional codes (CCs) of rate r/(r + 1). The CSF is derived by reformulating the parity-check constraints using the punctured dual codewords, and can be implemented either as a parity-check matrix (PCM) or as a multi-binary convolutional (MBC) structure. The former approach significantly reduces the complexity of the decoding process compared to conventional maximum a posteriori (MAP) decoding. Simulation results show that belief propagation (BP) decoding on the CSF achieves comparable performance to conventional MAP decoding, with a penalty of only 0.1 dB. Additionally MAP decoding on the MBC trellis yields identical performance. This enables a decoding complexity reduction ranging from 50% to 90% depending on the code rate. Furthermore, simulation results for turbo decoding shows around 0.3 dB degradation at lower rates and minimal impact (less than 0.15 dB) at higher rates compared to iterative MAP-based decoding.
In the context of in-memory computing, this paper investigates binary neural networks (BNNs) implemented with memristor crossbars. A key issue in these architectures is the difficulty in setting crossbar conductance values with an arbitrary precision, which introduces computational noise into the BNN layers. The paper first introduces a theoretical analysis of the impact of noise on the final computation, deriving an analytical expression for the error probability at the output of a BNN layer. This expression is validated through MonteCarlo simulations, demonstrating its accuracy in predicting error probabilities. Furthermore, the paper introduces a novel linear block code designed to mitigate the effects of noise in conductance values. This coding scheme is accompanied by a dedicated low-complexity belief propagation decoder that operates over the set of integers. Simulation results indicate a significant improvement in bit error rate for the proposed coding method compared to hard thresholding.
Error control coding for the systems for the previous generations (2G to 5G) has assumed the inputs bits to be uniformly distributed. However, some sources such as downlink (DL) HARQ-ACK payload are inherently non-uniformly distributed. Hence, there is room for significant performance improvement by employing joint source channel coding design aided by deep learning based techniques. We focus on AI/ML based codebook learning and decoder design for HARQ-ACK payload in order to exploit the source distribution in the control channel codebook and perform Unequal Error Protection (UEP) for the source bits encoded in the codewords. Our results demonstrate 4 - 8 dB reduction in the transmit power required to achieve the desired ACK and NACK error rates, compared to the 5G NR compliant encoder and decoder design.
Forward error correction (FEC) plays a vital role in optical communication. The open FEC code (FEC) is a very efficient code suitable for high-speed optical communication and used for e.g. 200-800 Gbit/s solutions. The oFEC code is based on component BCH codes. We investigate maximum a posteriori (MAP) algorithms for soft-in soft-out (SISO) decoding of the BCH codes and present an efficient algorithm that speeds up this MAP decoding significantly. In simulations, we demonstrate a SISO MAP decoding that satisfies the ITU-T 709.3 specification of a bit error rate (BER) output value of 10(-9) at a BER input value of 0.025. In terms of coding gain, this corresponds to an improvement of 0.4 dB compared to the requirement of the ITU-T 709.3 specification.
In this paper, we explore a construction of binary parity-check codes from nonbinary codes, with a specific focus on Reed-Solomon codes. The parity-check matrices of these codes come from the 1964 construction of superimposed codes by Kautz and Singleton. While some basic bounds on parameters of these parity-check codes are known, the specific parameters are highly dependent on the nonbinary code that one starts with. In this paper, we delve more deeply into parameters of this new class of girth six codes, showing that they are quasi-cyclic, providing bounds on the dimensions of the codes, and bounds on minimum distances in some specific cases. We present a table of small code parameters and note that some of these codes meet the best known minimum distance for binary codes. We draw connections to similar constructions in the literature, but importantly, while existing literature on related codes is largely simulation-based, we present a novel algebraic approach to determining new bounds on parameters of these codes.
This paper introduces a novel decomposed parallel concatenated convolutional code (DPCCC) structure designed to increase the minimum Hamming distance (mHD) and reduce the error floor compared to conventional PCCC structures, such as turbo codes (TCs). In DPCCC, the input frame is partitioned into subframes, each independently interleaved and encoded. This decomposition targets mHD codewords, partly through mitigating the quadratic increase in the multiplicity of periodic input weight-2 sequences, thus lowering their probability of co-occurrence in the component codes. We also propose a design algorithm that optimizes each subframe interleaver while accounting for inter-frame dependencies. Simulation results demonstrate that the proposed DPCCC structure, combined with the tailored interleaver design, achieves competitive performance compared to other code classes, with notable improvements in mHD and error floor performance over TCs, while maintaining rate compatibility and supporting low-latency decoder architectures.
In this paper, we investigate ways to mitigate the problem of decoder error propagation (DEP) in sliding window decoding (SWD) of protograph-based spatially coupled low- density parity-check (SC-LDPC) codes for large frame length or streaming applications. In particular, in order to avoid subdividing a long frame into a series of shorter frames by using termination to combat DEP, we consider altering the code design by introducing occasional doped symbols into the encoded sequence, where the doping is accomplished by fixing the values of all or some of the variable nodes (VNs) at certain positions in the protograph.An important practical consideration in many applications is the ability to use systematic encoding. This necessitates that no more than a fraction of the VNs at any given position can be doped, i.e., that fractional doping be employed. We begin by showing numerically that full doping of a single position in the protograph of a long frame improves performance relative to that of terminating the frame at half its length. We then show that fractional doping of consecutive VN positions at a given location in the protograph is comparable to full doping. We also show that spreading fractionally doped positions over multiple locations in the protograph results in additional gains, where the effective code rate of a doped frame is always at least as high as that of the shorter terminated frame.
