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At least 541 records · Page 30

Some Decomposable Codes: The |a + x|b + x|a + b + x| Construction

Codes with decomposable structure allow the use of multistage decoding procedures to achieve suboptimum bounded-distance error performance with reduced decoding complexity. This correspondence presents some new decomposable codes, including a class of distance-8 codes, that are constructed based on the|a + x|b + x|a + b + x| construction method. Some existing best codes are shown to be decomposable and hence can be decoded with multistage decoding.

Fossorier, Marc P. C.↗

Vector adaptive predictive coder for speech and audio

A real-time vector adaptive predictive coder which approximates each vector of K speech samples by using each of M fixed vectors in a first codebook to excite a time-varying synthesis filter and picking the vector that minimizes distortion. Predictive analysis for each frame determines parameters used for computing from vectors in the first codebook zero-state response vectors that are stored at the same address (index) in a second codebook. Encoding of input speech vectors s.sub.n is then carried out using the second codebook. When the vector that minimizes distortion is found, its index is transmitted to a decoder which has a codebook identical to the first codebook of the decoder. There the index is used to read out a vector that is used to synthesize an output speech vector s.sub.n. The parameters used in the encoder are quantized, for example by using a table, and the indices are transmitted to the decoder where they are decoded to specify transfer characteristics of filters used in producing the vector s.sub.n from the receiver codebook vector selected by the vector index transmitted.

Chen, Juin-Hwey↗

Constructing LDPC Codes from Loop-Free Encoding Modules

A method of constructing certain low-density parity-check (LDPC) codes by use of relatively simple loop-free coding modules has been developed. The subclasses of LDPC codes to which the method applies includes accumulate-repeat-accumulate (ARA) codes, accumulate-repeat-check-accumulate codes, and the codes described in Accumulate-Repeat-Accumulate-Accumulate Codes (NPO-41305), NASA Tech Briefs, Vol. 31, No. 9 (September 2007), page 90. All of the affected codes can be characterized as serial/parallel (hybrid) concatenations of such relatively simple modules as accumulators, repetition codes, differentiators, and punctured single-parity check codes. These are error-correcting codes suitable for use in a variety of wireless data-communication systems that include noisy channels. These codes can also be characterized as hybrid turbolike codes that have projected graph or protograph representations (for example see figure); these characteristics make it possible to design high-speed iterative decoders that utilize belief-propagation algorithms. The present method comprises two related submethods for constructing LDPC codes from simple loop-free modules with circulant permutations. The first submethod is an iterative encoding method based on the erasure-decoding algorithm. The computations required by this method are well organized because they involve a parity-check matrix having a block-circulant structure. The second submethod involves the use of block-circulant generator matrices. The encoders of this method are very similar to those of recursive convolutional codes. Some encoders according to this second submethod have been implemented in a small field-programmable gate array that operates at a speed of 100 megasymbols per second. By use of density evolution (a computational- simulation technique for analyzing performances of LDPC codes), it has been shown through some examples that as the block size goes to infinity, low iterative decoding thresholds close to channel capacity limits can be achieved for the codes of the type in question having low maximum variable node degrees. The decoding thresholds in these examples are lower than those of the best-known unstructured irregular LDPC codes constrained to have the same maximum node degrees. Furthermore, the present method enables the construction of codes of any desired rate with thresholds that stay uniformly close to their respective channel capacity thresholds.

Divsalar, Dariush↗

Method and System for Temporal Filtering in Video Compression Systems

Three related innovations combine improved non-linear motion estimation, video coding, and video compression. The first system comprises a method in which side information is generated using an adaptive, non-linear motion model. This method enables extrapolating and interpolating a visual signal, including determining the first motion vector between the first pixel position in a first image to a second pixel position in a second image; determining a second motion vector between the second pixel position in the second image and a third pixel position in a third image; determining a third motion vector between the first pixel position in the first image and the second pixel position in the second image, the second pixel position in the second image, and the third pixel position in the third image using a non-linear model; and determining a position of the fourth pixel in a fourth image based upon the third motion vector. For the video compression element, the video encoder has low computational complexity and high compression efficiency. The disclosed system comprises a video encoder and a decoder. The encoder converts the source frame into a space-frequency representation, estimates the conditional statistics of at least one vector of space-frequency coefficients with similar frequencies, and is conditioned on previously encoded data. It estimates an encoding rate based on the conditional statistics and applies a Slepian-Wolf code with the computed encoding rate. The method for decoding includes generating a side-information vector of frequency coefficients based on previously decoded source data and encoder statistics and previous reconstructions of the source frequency vector. It also performs Slepian-Wolf decoding of a source frequency vector based on the generated side-information and the Slepian-Wolf code bits. The video coding element includes receiving a first reference frame having a first pixel value at a first pixel position, a second reference frame having a second pixel value at a second pixel position, and a third reference frame having a third pixel value at a third pixel position. It determines a first motion vector between the first pixel position and the second pixel position, a second motion vector between the second pixel position and the third pixel position, and a fourth pixel value for a fourth frame based upon a linear or nonlinear combination of the first pixel value, the second pixel value, and the third pixel value. A stationary filtering process determines the estimated pixel values. The parameters of the filter may be predetermined constants.

Lu, Ligang↗

Method and system for efficient video compression with low-complexity encoder

Disclosed are a method and system for video compression, wherein the video encoder has low computational complexity and high compression efficiency. The disclosed system comprises a video encoder and a video decoder, wherein the method for encoding includes the steps of converting a source frame into a space-frequency representation; estimating conditional statistics of at least one vector of space-frequency coefficients; estimating encoding rates based on the said conditional statistics; and applying Slepian-Wolf codes with the said computed encoding rates. The preferred method for decoding includes the steps of; generating a side-information vector of frequency coefficients based on previously decoded source data, encoder statistics, and previous reconstructions of the source frequency vector; and performing Slepian-Wolf decoding of at least one source frequency vector based on the generated side-information, the Slepian-Wolf code bits and the encoder statistics.

Chen, Jun↗

Self-configurable radio receiver system and method for use with signals without prior knowledge of signal defining characteristics

A method, radio receiver, and system to autonomously receive and decode a plurality of signals having a variety of signal types without a priori knowledge of the defining characteristics of the signals is disclosed. The radio receiver is capable of receiving a signal of an unknown signal type and, by estimating one or more defining characteristics of the signal, determine the type of signal. The estimated defining characteristic(s) is/are utilized to enable the receiver to determine other defining characteristics. This in turn, enables the receiver, through multiple iterations, to make a maximum-likelihood (ML) estimate for each of the defining characteristics. After the type of signal is determined by its defining characteristics, the receiver selects an appropriate decoder from a plurality of decoders to decode the signal.

Hamkins, Jon↗

Encoding of symbols for a computer interconnect based on frequency of symbol values

Data are serially communicated over an interconnect between an encoder and a decoder. The encoder includes a first training unit to count a frequency of symbol values in symbol blocks of a set of N number of symbol blocks in an epoch. A circular shift unit of the encoder stores a set of most-recently-used (MRU) amplitude values. An XOR unit is coupled to the first training unit and the first circular shift unit as inputs and to the interconnect as output. A transmitter is coupled to the encoder XOR unit and the interconnect and thereby contemporaneously sends symbols and trains on the symbols. In a system, a device includes a receiver and decoder that receive, from the encoder, symbols over the interconnect. The decoder includes its own training unit for decoding the transmitted symbols.

SeyedzadehDelcheh, SeyedMohammad↗

Protograph based LDPC codes with minimum distance linearly growing with block size

We propose several LDPC code constructions that simultaneously achieve good threshold and error floor performance. Minimum distance is shown to grow linearly with block size (similar to regular codes of variable degree at least 3) by considering ensemble average weight enumerators. Our constructions are based on projected graph, or protograph, structures that support high-speed decoder implementations. As with irregular ensembles, our constructions are sensitive to the proportion of degree-2 variable nodes. A code with too few such nodes tends to have an iterative decoding threshold that is far from the capacity threshold. A code with too many such nodes tends to not exhibit a minimum distance that grows linearly in block length. In this paper we also show that precoding can be used to lower the threshold of regular LDPC codes. The decoding thresholds of the proposed codes, which have linearly increasing minimum distance in block size, outperform that of regular LDPC codes. Furthermore, a family of low to high rate codes, with thresholds that adhere closely to their respective channel capacity thresholds, is presented. Simulation results for a few example codes show that the proposed codes have low error floors as well as good threshold SNFt performance.

minimum decoding↗

Accumulate repeat accumulate codes

In this paper we propose an innovative channel coding scheme called 'Accumulate Repeat Accumulate codes' (ARA). This class of codes can be viewed as serial turbo-like codes, or as a subclass of Low Density Parity Check (LDPC) codes, thus belief propagation can be used for iterative decoding of ARA codes on a graph. The structure of encoder for this class can be viewed as precoded Repeat Accumulate (RA) code or as precoded Irregular Repeat Accumulate (IRA) code, where simply an accumulator is chosen as a precoder. Thus ARA codes have simple, and very fast encoder structure when they representing LDPC codes. Based on density evolution for LDPC codes through some examples for ARA codes, we show that for maximum variable node degree 5 a minimum bit SNR as low as 0.08 dB from channel capacity for rate 1/2 can be achieved as the block size goes to infinity. Thus based on fixed low maximum variable node degree, its threshold outperforms not only the RA and IRA codes but also the best known LDPC codes with the dame maximum node degree. Furthermore by puncturing the accumulators any desired high rate codes close to code rate 1 can be obtained with thresholds that stay close to the channel capacity thresholds uniformly. Iterative decoding simulation results are provided. The ARA codes also have projected graph or protograph representation that allows for high speed decoder implementation.

Low Density Parity Check codes (LDPC)↗

NuGraph2 with context-aware inputs: physics-inspired improvements in semantic segmentation

Graph neural networks have recently shown strong promise for event reconstruction tasks in Liquid Argon Time Projection Chambers, yet their performance remains limited for underrepresented classes of particles, such as Michel electrons. In this work, we investigate physics-informed strategies to improve semantic segmentation within the NuGraph2 architecture. We explore three complementary approaches: (i) enriching the input representation with context-aware features derived from detector geometry and track continuity, (ii) introducing auxiliary decoders to capture class-level correlations, and (iii) incorporating energy-based regularization terms motivated by Michel electron energy distributions. Experiments on MicroBooNE public datasets show that physics-inspired feature augmentation yields the largest gains, particularly boosting Michel electron precision and recall by disentangling overlapping latent space regions. In contrast, auxiliary decoders and energy-regularization terms provided limited improvements, partly due to the hit-level nature of NuGraph2, which lacks explicit particle- or event-level representations. Our findings highlight that embedding physics context directly into node-level inputs is more effective than imposing task-specific auxiliary losses, and suggest that future hierarchical architectures such as NuGraph3, with explicit particle- and event-level reasoning, will provide a more natural setting for advanced decoders and physics-based regularization. The code for this work is publicly available on Github at https://github.com/vitorgrizzi/nugraph_phys/tree/main_phys.

Other Experiments↗

Learning PDFs through interpretable latent representations in Mellin space

Representing the parton distribution functions (PDFs) of the proton and other hadrons through flexible, high-fidelity parametrizations has been a long-standing goal of particle physics phenomenology. This is particularly true since the chosen parametrization methodology can play an influential role in the ultimate PDF uncertainties as extracted in QCD global analyses; these, in turn, are often determinative of the reach of experiments at the LHC and other facilities to nonstandard physics, including at large 𝑥, where parametrization effects can be significant. In this study, we explore a series of encoder-decoder machine-learning (ML) models with various neural-network topologies as efficient means of reconstructing PDFs from meaningful information stored in an interpretable latent space. Given recent effort to pioneer synergies between QCD analyses and lattice-gauge calculations, we formulate a latent representation based on the behavior of PDFs in Mellin space, i.e., their integrated moments, and test the ability of various models to decode PDFs from this information faithfully. We introduce a numerical package, PDFdecoder, which implements several encoder-decoder models to reconstruct PDFs with high fidelity and use this end-to-end tool to explore how such neural-network-based models might connect PDF parametrizations to underlying properties like their Mellin moments. We additionally dissect patterns of learned correlations between encoded Mellin moments and reconstructed PDFs that suggest opportunities for further improvements to ML-based approaches to PDF parametrizations and uncertainty quantification.

Machine learning↗

Leveraging Qubit Loss Detection in Fault-Tolerant Quantum Algorithms

Qubit loss errors constitute a dominant source of noise in many quantum hardware systems, particularly in neutral-atom quantum computers. We develop a theoretical framework to effectively detect and correct loss errors in logical algorithms and leverage such loss information in decoding. Considering general quantum error correction codes and logical circuits, we introduce a delayed-erasure decoder for experimentally motivated error models which leverages information from delayed loss detection to accurately correct loss errors, even when the precise moment of the error is unknown. Using this decoder, we identify strategies for detecting and correcting loss errors based on the logical circuit structure. For deep circuits prior to logical measurement, we explore methods to integrate loss detection into syndrome extraction with minimal overhead, identifying optimal strategies depending on the qubit loss fraction in the noise and hardware capabilities. In contrast, we find that many key algorithmic subroutines involve frequent gate teleportation, shortening the circuit depth before logical measurement and naturally replacing qubits with no additional experimental overhead. We simulate this setting using a toy model algorithm for small-angle synthesis and find a significant performance improvement as the loss fraction increases. These results provide a path forward for advancing large-scale fault-tolerant quantum computation in systems with loss error detection.

atoms↗

HydraGNN_Predictive_GFM_2026 - Ensemble of predictive graph foundation models for atomistic materials modeling

This release contains data and parameters of HydraGNN-based graph foundation models trained as a result of the work published in the pre-print "Exascale Multi-Task Graph Foundation Models for Imbalanced, Multi-Fidelity Atomistic Data" by M. Lupo Pasini et al. (https://arxiv.org/abs/2604.15380). We jointly train on 16 open first-principles datasets (544+ million structures covering 85+ elements) using a multi-task architecture with per-dataset heads and a scalable ADIOS2/DDStore data pipeline. On Frontier, we execute six large-scale DeepHyper hyperparameter optimization campaigns in FP64 and promote the top-performing message-passing models to sustained 2,048-node training, yielding a PaiNN-based lead model. The version of HydraGNN used to generate the outputs provided in this release is HydraGNN v5.0 (https://github.com/ORNL/HydraGNN/releases/tag/v5.0) The list of datasets used for the training of the graph foundation model is the following: 1) Alexandria [1] 2) ANI1x [2] 3) MPTrj [3] 4) Open Catalyst 2020 (OC20) [4] 5) Open Catalyst 2022 (OC22) [5] 6) Open Catalyst 2025 (OC25) [6] 7) Open Direct ir Capture 2023 (ODAC23) [7] 8) Open Materials 2024 (OMat24) [8] 9) Open Molecules 2025 (OMol25) [9] 10) OMol25-neutral (subset of OMol25 that contains only molecules with zero total charge) 11) OMol25-non-neutral (subset of OMol25 that contains only molecules with non-zero total charge) 12) Open Polymers 2026 (OPoly2026) [10] 13) Nabla2DFT [11] 14) QCML [12] 15) QM7X [reference 13] 16) transition1x [14] Dataset references: [1] J. Schmidt et al., “A dataset of 175k stable and metastable materials calculated with the PBEsol and SCAN functionals,” Scientific Data, vol. 9, p. 64, 2022. [2] J. S. Smith et al., “The ANI-1ccx and ANI-1x data sets, coupled-cluster and density functional theory properties for molecules,” Scientific Data, vol. 7, p. 134, 2020. [Online]. Available: https: //www.nature.com/articles/s41597-020-0473-z [3] A. Jain et al., “Commentary: The Materials Project: A materials genome approach to accelerating materials innovation,” APL Materials, vol. 1, no. 1, p. 011002, 07 2013. [Online]. Available: https://doi.org/10.1063/1.4812323 [4] L. Chanussot et al., “Open catalyst 2020 (oc20) dataset and community challenges,” ACS Catalysis, vol. 11, no. 10, pp. 6059–6072, 2021. [Online]. Available: https://doi.org/10.1021/acscatal.0c04525 [5] K. Tran et al., “Open catalyst 2022 (oc22) dataset and challenges for oxidation electrocatalysts,” ACS Catalysis, vol. 13, no. 5, pp. 3066–3084, 2023. [Online]. Available: https://doi.org/10.1021/acscatal.2c05426 [6] S. J. Sahoo et al., “The open catalyst 2025 (oc25) dataset and models for solid-liquid interfaces,” arXiv preprint arXiv:2509.17862, 2025. [Online]. Available: https://arxiv.org/abs/2509.17862 [7] A. Sriram et al., “The open DAC 2023 dataset and challenges for sorbent discovery in direct air capture,” ACS Central Science, vol. 10, no. 5, pp. 923–941, 2024. [8] L. Barroso-Luque et al., “Open materials 2024 (omat24) inorganic materials dataset and models,” 2024. [Online]. Available: https://arxiv.org/abs/2410.12771 [9] D. S. Levine et al., “The open molecules 2025 (OMol25) dataset, evaluations, and models,” 2025. [Online]. Available: https://arxiv.org/abs/2505.08762 [10] D. S. Levine et al., The open polymers 2026 (OPoly26) dataset and evaluations,” arXiv preprint arXiv:2512.23117, 2025. [Online]. Available: https://arxiv.org/abs/2512.23117 [11] K. Khrabrov et al., “Nabla2dft: A universal quantum chemistry dataset of drug-like molecules and a benchmark for neural network potentials,” in NeurIPS 2024 Datasets and Benchmarks Track, 2024. [Online]. Available: https://openreview.net/forum?id=ElUrNM9U8c [12] S. Ganscha et al., “The QCML dataset, quantum chemistry reference data from 33.5M DFT and 14.7B semi-empirical calculations,” Scientific Data, vol. 12, p. 406, 2025. [13] J. Hoja et al., “QM7-X, a comprehensive dataset of quantum-mechanical properties spanning the chemical space of small organic molecules,” Scientific Data, vol. 8, p. 43, 2021. [Online]. Available: https://www.nature.com/articles/s41597-021-00812-2 [14] M. Schreiner et al., “Transition1x - a dataset for building generalizable reactive machine learning potentials,” Scientific Data, vol. 9, p. 779, 2022. The folder "datasets_ADIOS2_format" contains the set of pre-processed datasets in Adaptable I/O System (ADIOS) format (https://www.exascaleproject.org/research-project/adios/) that have been used for the development and training of GFMs in this work. The "datasets_ADIOS2_format" directory contains 2 sub-directories, one for the version "v1" of the datasets and one for the version "v2" of the datasets. The version "v1" of the datasets provides values of the total energy as they are extracted from the original data as it was released by the respective institutions. The version "v2" of the datasets provides values of the energy that have been realigned. The realignment was performed by training a linear regression model that predicts the total energy as a function of the chemical composition of the atomistic structure, and then subtract such prediction from the original value of the total energy. Both folders "v1" and "v2" contain 16 sub-directories, each corresponding to an ADIOS2-formatted dataset The folder "DeepHyper-results" contains the configurational files and model's parameters for all the 186 HPO trials that were successfully completed by the scalable hyperparameter optimization (HPO) runs on Frontier. The content of the folder "DeepHyper-results" I structured as follows: 1) task-list.txt: list of mpnn name, jobid, and deephyper task id 2) gfm_${MPNN}_${JOBID}_0.${TASKID}: run directory with checkpoint files 3) gfm_${MPNN}: deephyper summary directory (*.csv) for each specific MPNN type 4) deephyper-experiment-${JOBID}: output and error logs for each job The file "deephyper-sorted.csv" contains the details of each HydraGNN model built and tested by HPO, obtained by merging the (*.csv) filed from each HPO run executed. Out of all the HPO trials, we selected 10 to continue the training of the respective HydraGNN models. Due to limited computational budget available in the LRN070 allocation we could not complete the training till convergence for all these 10 selected models. The folder "models" contains multiple sub-folders, one per each HydraGNN model trained. Each model sub-folder contains the parameters of each HydraGNN model, with multiple checkpoint-restarts. The list of sub-folders are as follows: 1) multidataset_hpo-BEST1-fp64 2) multidataset_hpo-BEST2-fp64 3) multidataset_hpo-BEST3-fp64 4) multidataset_hpo-BEST4-fp64 5) multidataset_hpo-BEST5-fp64 6) multidataset_hpo-BEST6-fp64 7) multidataset_hpo-BEST7-fp64 8) multidataset_hpo-BEST8-fp64 9) multidataset_hpo-BEST9-fp64 10) multidataset_hpo-BEST10-fp64 Within each one of these folders, additional auxiliary log files are provided with descriptions about how the training proceeded. The lead PaiNN-model is contained inside "multidataset_hpo-BEST6-fp64". The file "mlp_branch_weights" contains the parameters of the multi-layer perceptron (MLP) used to reconcile the predictions of the 16 output decoding heads of the HydragNN architectures. The MLP takes in input the chemical composition of the atomistic structure and predicts averaging weights to linearly mix the predictions of each output decoding head toward consolidating them into a single one. The folder "1.1billion-structure-inference" contains 1.1 billion atomistic structures randomly generated. Each structures is associated with energy and forces predicted with the lead-PaiNN model combined with the MLP model for reconciliation of the multi-branch predictions generated by the 16 output decoding heads. The folder "1.1billion-structure-inference" contains 9,300 (*.tar.gz) subdirectories, one per Frontier compute node used to execute the inference at exascale. Once uncompressed, each (*.tar.gz) subdirectory contains an ADIOS2 (*.bp) file container, where each atomistic structure is stored as a PyTorch-Geometric Data object. The file "export_dataset_environment_variables.sh" contains the environment variables that need to be set before running the HydraGNN code to reproduce the results provided in this dataset release. The code that can be used to load the ADIOS2 files, load HydraGNN models, and run inference is available at: https://github.com/ORNL/HydraGNN/releases/tag/v5.0

36 MATERIALS SCIENCE↗

Unique frequency-shift-keyed demodulation system

Frequency-Shift-Keyed /FSK/ demodulator provides a frequency discriminator whose outputs are separate and applied to two identical decoding channels, one decoding binary ones and the other decoding binary zeros. This demodulator rejects data applied to it at any frequency higher than design.

Staloff, C.↗

Conceptual design of a 10 to the 8th power bit magnetic bubble domain mass storage unit and fabrication, test and delivery of a feasibility model

The conceptual design of a highly reliable 10 to the 8th power-bit bubble domain memory for the space program is described. The memory has random access to blocks of closed-loop shift registers, and utilizes self-contained bubble domain chips with on-chip decoding. Trade-off studies show that the highest reliability and lowest power dissipation is obtained when the memory is organized on a bit-per-chip basis. The final design has 800 bits/register, 128 registers/chip, 16 chips/plane, and 112 planes, of which only seven are activated at a time. A word has 64 data bits +32 checkbits, used in a 16-adjacent code to provide correction of any combination of errors in one plane. 100 KHz maximum rotational frequency keeps power low (equal to or less than, 25 watts) and also allows asynchronous operation. Data rate is 6.4 megabits/sec, access time is 200 msec to an 800-word block and an additional 4 msec (average) to a word. The fabrication and operation are also described for a 64-bit bubble domain memory chip designed to test the concept of on-chip magnetic decoding. Access to one of the chip's four shift registers for the read, write, and clear functions is by means of bubble domain decoders utilizing the interaction between a conductor line and a bubble.

Source record↗

Variable-length codes and the Fano metric.

It is shown that the metric proposed originally by Fano for sequential decoding is precisely the required statistic for minimum-error-probability decoding of variable-length codes. The analysis shows further that the 'natural' choice of bias in the metric is the code rate and gives insight into why the Fano metric has proved to be the best practical choice in sequential decoding. The recently devised Jelinek-Zigangirov 'stack algorithm' is shown to be a natural consequence of this interpretation of the Fano metric. Finally, it is shown that the elimination of the bias in the 'truncated' portion of the code tree gives a slight reduction in average computation at the sacrifice of increased error probability.

Massey, J. L.↗

Performance evaluation of a class of systematic, rate (M-1)/M, convolutional codes

The implementation and performance evaluation are described for a class of rate (M-1)/M, systematic, convolutional codes being decoded with a simple majority logic decoder. The encoding logic appends one parity bit for each PCM telemetry word. It is shown that over the critical range of received PCM telemetry signal-to-noise ratios, this coding procedure produces a net coding gain of from 1.5 to 2.5 db relative to an equal power transmission of uncoded PCM telemetry. Being a low-redundancy systematic code, it is possible to process this data without convolutional decoding with a small rate loss penalty of about 0.5 db.

Greene, E. P.↗

Free distance bounds for convolutional codes

The best asymptotic bounds presently known on free distance for convolutional codes are presented from a unified point of view. Upper and lower bounds for both time-varying and fixed codes are obtained. A comparison is made between bounds for nonsystematic and systematic codes which shows that more free distance is available with nonsystematic codes. This result is important when selecting codes for use with sequential or maximum-likelihood (Viterbi) decoding since the probability of decoding error is closely related to the free distance of the code. An ancillary result, used in proving the lower bound on free distance for time-varying nonsystematic codes, furnishes a generalization of two earlier bounds on the definite decoding minimum distance of convolutional codes.

Costello, D. J., Jr.↗