Modified OSD Algorithm With Reduced Gaussian Elimination

The conventional soft decision decoding (SDD) methods require various hard decision decoders (HDDs) based on different codes or re-manipulate the generator matrix by the complicated Gaussian elimination technique according to the bit reliability. This paper presents a general multi-class neural network (NN)-based decoder for the short linear block codes, where no HDD and Gaussian elimination are required once the NN is constructed. This network architecture performs multi-classification to select the messages with high occurrence probabilities and chooses the best codeword on a maximum likelihood basis. Simulation results show that the developed approach outperforms the existing deep neural network (DNN)-based decoders in terms of decoding time and bit error rate (BER). The error-correcting performance is also superior to the conventional Chase-II algorithm and is close to the ordered statistics decoding (OSD) in most cases. For Bose–Chaudhuri–Hocquenghem (BCH) codes, the SNR is improved by 1dB to 4dB as the BER is 10−4. For the (23, 12) quadratic residue (QR) code, the SNR is improved by 2dB when the BER is 10−3. The developed NN-based decoder is quite general and applicable to various short linear block codes with good BER performance.

Data center networks (DCNs) require a low-cost, low-power optical transceiver to handle increased traffic from generative artificial intelligence, video streaming services, and more. Improving the required signal-to-noise ratio (RSNR) by digital signal processing such as forward error correction (FEC) mitigates the requirements for electrical and optical components. The optical transceivers in DCNs exploit a low-complexity soft-decision (SD) FEC, consisting of short block-length linear error-correcting codes and a low-complexity SD decoder (SDD), such as Chase decoding and ordered statistics decoding. The low-complexity SDD efficiently approaches a maximum likelihood decoding (MLD). However, the decoding performance of MLD is limited by its finite block length. In this paper, we describe the details of our proposed channel-polarized multilevel coding with iterative decoding (CP-MLC-ID). The proposed CP-MLC-ID improves the decoding performance by extending the codeword length to weakly and indirectly connect codewords via bypassed bits. The 19.5% –OH CP-MLC-ID using 128-bit extended Bose–Chaudhuri–Hocquenghem (eBCH) and KP4 codes outperforms the concatenated eBCH and KP4 codes with a net coding gain of 0.25 and 0.40dB for the same and double the number of SDDs, respectively. We also investigate the dependency of the decoding performance on the size of a bit interleaver. The performance degradation of CP-MLC-ID using an 8-bit interleaver is about 0.1dB compared to using the large-bit interleaver. Our results indicate that even a weak connection by exclusive-OR between codewords improves the decoding performance, compared to simple concatenated codes in the DCNs.

This paper studies linear and affine error-correcting codes for correcting synchronization errors such as insertions and deletions. We call such codes linear/affine insdel codes. Linear codes that can correct even a single deletion are limited to having an information rate at most (achieved by the trivial two fold repetition code). Previously, it was (erroneously) reported that more generally no nontrivial linear codes correcting deletions exist, i.e., that the -fold repetition codes and its rate of are basically optimal for any . We disprove this and show the existence of binary linear codes of length and rate just below capable of correcting insertions and deletions. This identifies rate as a sharp threshold for recovery from deletions for linear codes and reopens the quest for a better understanding of the capabilities of linear codes for correcting insertions/deletions. We prove novel outer bounds and existential inner bounds for the rate vs. (edit) distance trade-off of linear insdel codes. We complement our existential results with an efficient synchronization-string-based transformation that converts any asymptotically good linear code for Hamming errors into an asymptotically good linear code for insdel errors. Last, we show that the -rate limitation does not hold for affine codes by giving an explicit affine code of rate which can efficiently correct a constant fraction of insdel errors.

In this contribution, KNIH algorithm and Ordered Statistic Decoding (OSD) algorithm are investigated, based on which, a novel KNIH-OSD parallel decoding scheme for LDPC codes is proposed. KNIH decoding algorithm processes certain LRPs of a received sequence, while OSD algorithm processes certain MRIPs of a received sequence. If there are too many errors in MRIPs, the OSD algorithm will fail, on the contrary, if there are too many errors in LRPs, the KNIH algorithm will fail. This contribution proposes a parallel decoding scheme based on the complementary characteristic of these two algorithms. Simulation results show that the proposed scheme is feasible and effective. Compared with OSD and KNIH algorithm, the decoding performance is improved.

In this work, the homomorphism of the classic linear block code in linear network coding for the case of binary field and its extensions is studied. It is proved that the classic linear error-control block code is homomorphic network error-control code in network coding. That is, if the source packets at the source node for a linear network coding are precoded using a linear block code, then every packet flowing in the network regarding to the source satisfies the same constraints as the source. As a consequence, error detection and correction can be performed at every intermediate nodes of multicast flow, rather than only at the destination node in the conventional way, which can help to identify and correct errors timely at the error-corrupted link and save the cost of forwarding error-corrupted data to the destination node when the intermediate nodes are ignorant of the errors. In addition, three examples are demonstrated which show that homomorphic linear code can be combined with homomorphic signature, McEliece public-key cryptosystem and unequal error protection respectively and thus have a great potential of practical utility.

Trellis diagrams have been traditionally exploited for decoding of convolutional codes [197]. Trellis design procedures for linear block codes have been under investigation since 1974 when Bahl, Cocke, Jelinek and Raviv introduced a method of representing the words in an arbitrary linear block code by the path labels in the trellis [5]. It has been found by Wolf J. [204] that an arbitrary linear (n, k,d min ) block code over GF(q), q ≥ 2, can be represented by its trellis diagram which contains N c = n + 1 columns (or depths) and N s ≤ q min {k,n-k} states. He found that such a trellis, also known as a syndrome or a Wolf’s trellis, could be constructed for the soft maximum likelihood trellis decoding (SMLTD) of a code. However, syndrome trellises have a non-regular structure (except for some particular codes, such as cyclic codes) that complicates their practical implementation.

Irregular LDPC codes can achieve better error rate performance than regular LDPC codes. However, irregular LDPC codes have higher error floors than regular LDPC codes. The Ordered Statistic Decoding (OSD) algorithm achieves approximate Maximum Likelihood (ML) decoding. ML decoding is effective to lower error floors. However, the OSD estimates satisfy the parity check equation of the LDPC code even the estimates are wrong. Hybrid decoder combining LLR-BP decoding algorithm and the OSD algorithm cannot also lower error floors, because wrong estimates also satisfy the LDPC parity check equation. We proposed the concatenated code constructed with an inner irregular LDPC code and an outer Cyclic Redundancy Check (CRC). Owing to CRC, we can detect wrong codewords from OSD estimates. Our CRC-LDPC code with hybrid decoder can lower error floors in an AWGN channel. In wireless communications, we cannot neglect the effects of the channel. The OSD algorithm needs the ordering of each bit based on the reliability. The Channel State Information (CSI) is used for deciding reliability of each bit. In this paper, we evaluate the Block Error Rate (BLER) of the CRC-LDPC code with hybrid decoder in a fast fading channel with perfect and imperfect CSIs where 'imperfect CSI' means that the distribution of channel and those statistical average of the fading amplitudes are known at the receiver. By computer simulation, we show that the CRC-LDPC code with hybrid decoder can lower error floors than the conventional LDPC code with hybrid decoder in the fast fading channel with perfect and imperfect CSIs. We also show that combining error detection with the OSD algorithm is effective not only for lowering the error floor but also for reducing computational complexity of the OSD algorithm.

Nonlinear channel codes are employed to protect two information fields that are related to the daylight saving time (DST) and leap second notifications within the enhanced WWVB broadcast format. The general codebook design criteria and the construction of these codes using integer linear program (ILP) is formulated. Specifically, two different variations are presented: i) minimizing the union bound of probability of word error (PWE), which is a tighter bound but has high computational complexity and ii) minimizing an analytical closed form approximation to the PWE, which has much lower complexity but is a looser bound on the exact PWE. By leveraging the highly non-uniform a-priori probabilities of the different messages being transmitted in these two fields, the proposed nonlinear codes provide coding gains of about 2–3 dB in the presence of additive white Gaussian noise (AWGN). When compared to checksum codes with the same rate, the nonlinear codes offer performance advantages of about 1–2 dB for the DST notification field. In the presence of impulsive noise, simulations and analysis show that the probability of word error when employing the proposed nonlinear code is lower by many orders of magnitude compared with that of the uncoded system.

A new family of nonlinear codes, called weak flip codes, is presented and is shown to contain many beautiful properties. In particular, the subfamily of fair weak flip codes can be seen as a generalization of linear codes. Different from linear codes that only exist for a number of codewords M being an integer-power of 2, the fair weak flip code can be defined for an arbitrary M. It is then noted that the fair weak flip codes are related to binary nonlinear Hadamard codes: both code families maximize the minimum Hamming distance and meet the Plotkin bound. However, while the binary nonlinear Hadamard codes have only been shown to possess good Hamming-distance properties, the fair weak flip codes are proven to be globally optimal (in the sense of minimizing the error probability) among all linear or nonlinear codes for the binary erasure channel (BEC) for many values of the blocklength n and for M≤6. For M>6, similar optimality results are conjectured. The results in this paper are founded on a new powerful tool for the analysis and generation of block codes: the column-wise approach to the codebook matrix.

Guessing Random Additive Noise Decoding (GRAND) excels at decoding high-rate codes but struggles to decode low-rate codes with reasonable complexity. Ordered Statistics Decoding (OSD) specifically excels in decoding short codes irrespective of rates; however, OSD necessitates the use of Gaussian elimination which introduces additional time, space and computational complexity. Partial Ordered Statistics Decoding (POSD) was proposed to reduce the time, space, and computational complexity of OSD; however, the current partition-based POSD has poor decoding performance since it does not generate test error patterns across partitions. In this paper, we propose to improve the decoding performance of POSD by incorporating test error patterns inspired by GRAND methods. This work offers a trade-off between performance and complexity compared to existing decoders such as GRAND and OSD. We enhance POSD by optimizing the scheduling of Test Error Patterns (TEPs) and show that our technique can be applied to any code in a standard form. At a target BER 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−4</sup> with eBCH (128,64) the enhanced error patterns achieve more than 0.6 dB gain in performance compared to the POSD with partition-based error patterns. Moreover, at a target frame error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−5</sup> , POSD uses 10× less binary operations compared to GRAND when decoding eBCH (128,64) and RLC(128,64) codes. With BCH (127,29) and RLC(128,32), at a target frame error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−2</sup> , POSD with enhanced error patterns with a maximum number of queries (MQ) of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> achieves up to a 2 dB gain to its GRAND equivalent which is using 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7</sup> maximum number of queries.

Linear block codes are studied for improving the reliability of message storage in computer memory with stuck-at defects and noise. The case when the side information about the state of the defects is available to the decoder or to the encoder is considered. In the former case, stuck-at cells act as erasures so that techniques for decoding linear block codes for erasures and errors can be directly applied. We concentrate on the complimentary problem of incorporating stuck-at information in the encoding of linear block codes. An algebraic model for stuck-at defects and additive errors is presented. The notion of a "partitioned" linear block code is introduced to mask defects known at the encoder and to correct random errors at the decoder. The defect and error correction capability of partitioned linear block codes is characterized in terms of minimum distances. A class of partitioned cyclic codes is introduced. A BCH-type bound for these cyclic codes is derived and employed to construct partitioned linear block codes with specified bounds on the minimum distances. Finally, a probabilistic model for the generation of stuck-at cells is presented. It is shown that partitioned linear block codes achieve the Shannon capacity for a computer memory with symmetric defects and errors.

This paper finds the best binary linear block codes to use when an IID uniform source is uniform scalar quantized for transmission over a binary symmetric channel, with performance measured by mean squared error (MSE). The best code is found by optimizing the source-channel rate allocation, the linear code, and the assignment of its codewords to quantization indices. A table shows the maximum attained SNR, together with a code that achieves it, for a range of channel transition probabilities and transmission rates. When optimized, the performance of such systems comes within 1 ~ 2 dB of that of the best nonlinear codes found by an iterative algorithm.

Finding the minimum distance of linear codes is in general a NP-hard problem, we propose an efficient algorithm to attack this problem. The principle of this approach is to search code words locally around the all-zero code word perturbed by a level of noise magnitude, in other words the maximum of noise that can be corrected by a Soft-In decoder, anticipating that the resultant nearest non-zero code words will most likely contain the minimum Hamming weight code word, whose Hamming weight is equal to the minimum distance of the linear code. A numerous results prove that the proposed algorithm is valid for general linear codes and it is very fast comparing to all others known techniques, therefore it is a good tool for computing. Comparing to Joanna's works, we proof that our algorithm has a low complexity with a fast time of execution. For some linear RQs, QDCs and BCHs codes with unknown minimum distance, we give a good estimation (true) of the minimum distance where the length is less than 439.

A fundamental result dating to Ligero (Des. Codes Cryptogr. '23) establishes that each fixed linear block code exhibits proximity gaps with respect to the collection of affine subspaces, in the sense that each given subspace either resides entirely close to the code, or else contains only a small portion which resides close to the code. In particular, any given subspace's failure to reside entirely close to the code is necessarily witnessed, with high probability, by a uniformly randomly sampled element of that subspace. We investigate a variant of this phenomenon in which the witness is not sampled uniformly from the subspace, but rather from a much smaller subset of it. We show that a logarithmic number of random field elements (in the dimension of the subspace) suffice to effect an analogous proximity test, with moreover only a logarithmic (multiplicative) loss in the possible prevalence of false witnesses. We discuss applications to recent noninteractive proofs based on linear codes, including Brakedown (CRYPTO '23).

Decoding a redundant code, when redefined as a means for estimating a posterior probability distribution given a prior one, has a simple output if one looks for the posterior distribution in a form separable with respect to the information symbols. The optimal distribution of this form results from minimizing its cross-entropy with respect to the prior distribution, which for a linear code leads to a system of implicit analog equations to be solved. This can be done by iteration, but the system has multiple solutions so that determining proper initial values becomes the main difficulty. We interpret any linear code as a kind of product of its parity checks whose successive decodings provide adequate initial values for solving the system, provided the received symbols are reordered by decreasing reliability (except for few inversions in order to keep the generator matrix in systematic form). Simulation results are presented.