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New Efficient Decoder of Linear Block Codes Based on Information Sets, Hash Techniques, and Machine Learning Models

Mohammed El Assad, Saïd Nouh, Seddiq El Kasmi Alaoui, Mohamed Azzouazi

Published Jul 30, 2026• 0 citations
Publisher

Abstract

Error-correcting codes improve the reliability of data exchanged between transmitters and receivers; achieving efficient and fast decoding is a major challenge. In this work, we propose an efficient, fast, and high-performing decoder named FastSyndISD. The first version of the proposed decoder uses hashing techniques to quickly find the information set that allows for error correction directly from the syndrome of the received word. Hashing information sets significantly reduce space complexity, instead of all correctable errors, compared to HSDec, LRDec, and EL-BoostDec. The second version utilizes a machine learning model (decision tree) to quickly find the information set that enables error correction directly from the syndrome of the received word. This second version does not require storing a hash table, since the information set is determined by a decision tree classifier model. The performance of the proposed decoder FastSyndISD is very encouraging regarding its bit error rate (BER). Indeed, it succeeds in correcting all errors whose weights are equal to or less than the correction capacity of the analyzed codes. Furthermore, a comparison with various other decoders highlights its effectiveness. A detailed study of its space complexity, its time complexity, and the factors influencing its performance has also been conducted, aiming to achieve an optimal BER while reducing computational time and memory usage. This comparison proves the significant success of the proposed decoder.

DOI
10.3390/info17080739

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•Jan 1, 1994•0 citations
Error-correcting codes improve the reliability of data exchanged between transmitters and receivers; achieving efficient and fast decoding is a major challenge. In this work, we propose an efficient, fast, and high-performing decoder named FastSyndISD. The first version of the proposed decoder uses hashing techniques to quickly find the information set that allows for error correction directly from the syndrome of the received word. Hashing information sets significantly reduce space complexity, instead of all correctable errors, compared to HSDec, LRDec, and EL-BoostDec. The second version utilizes a machine learning model (decision tree) to quickly find the information set that enables error correction directly from the syndrome of the received word. This second version does not require storing a hash table, since the information set is determined by a decision tree classifier model. The performance of the proposed decoder FastSyndISD is very encouraging regarding its bit error rate (BER). Indeed, it succeeds in correcting all errors whose weights are equal to or less than the correction capacity of the analyzed codes. Furthermore, a comparison with various other decoders highlights its effectiveness. A detailed study of its space complexity, its time complexity, and the factors influencing its performance has also been conducted, aiming to achieve an optimal BER while reducing computational time and memory usage. This comparison proves the significant success of the proposed decoder.
Error-correcting codes are used to partially or completely correct errors as much as possible, while ensuring high transmission speeds. Several Machine Learning (ML) models such as Logistic Regression and Decision tree have been applied to correct transmission errors. Among the most powerful ML techniques are aggregation methods which have yielded to excellent results in many areas of research. It is this excellence that has prompted us to consider their application for the hard decoding problem. In this sense, we have successfully designed, tested and validated our proposed EL-BoostDec decoder (hard decision decoder based on Ensemble Learning - Boosting technique) which is based on computing of the syndrome of the received word and on using Ensemble Learning techniques to find the corresponding corrigible error. The obtained results with EL-BoostDec are very encouraging in terms of the binary error rate (BER) that it offers. Practically EL-BoostDec has succeed to correct 100% of errors that have weights less than or equal to the correction capacity of studied codes. The comparison of EL-BoostDec with many competitors proves its power. A study of parameters which impact on EL-BoostDec performances has been established to obtain a good BER with minimum run time complexity.
Facing the challenge of enormous data sets variety, several machine learning-based algorithms for prediction (e.g, Support vector machine, multi layer perceptron and logistic regression) have been highly proposed and used over the last years in many fields. Error correcting codes (ECCs) are extensively used in practice to protect data against damaged data storage systems and against random errors due to noise effects. In this paper, we will use machine learning methods, especially multi-class logistic regression combined with the famous syndrome decoding algorithm. The main idea behind our decoding method which we call logistic regression decoder (LRDec) is to use the efficient multi-class logistic regression models to find errors from syndromes in linear codes such as bose, ray-chaudhuri and hocquenghem (BCH), and the quadratic residue (QR). Obtained results of the proposed decoder have a significant benefit in terms of bit error rate (BER) for random binary codes. The comparison of our decoder with many competitors proves its power. The proposed decoder has reached a success percentage of 100% for correctable errors in the studied codes.
This paper presents an approach to soft-decision decoding for a (n, k) binary linear block code. This approach achieves near-optimal to optimal performance for any signal-to-noise ratio value. The proposed decoding is based on the reordering of the received symbols according to their reliability measure and can be applied to any binary linear code. The basic idea is to generate a sequence of codewords such that there will be a high probability that the next codeword is closer to the received vector. In this algorithm the information provided by the soft-decisions is used to generate information sets utilized in decoding. A fast procedure checks whether a given set of k positions is an information set. A sequence of codewords is reconstructed using a sequence of information sets. Each codeword reconstructed is compared with the received word and the closest one is selected. Hence, the decoding algorithm is an iterative procedure designed to enhance the performance of the usual soft-decision decoding, since with each iteration of the procedure, real vectors successively closer to the received vector are examined. A high performance stop rule is utilized to reduce the number of generated information sets, stopping the decoding process when a codeword is closest to the received sequence.
In this correspondence, various aspects of reliability-based syndrome decoding of binary codes are investigated. First, it is shown that the least reliable basis (LRB) and the most reliable basis (MRB) are dual of each other. By exploiting this duality, an algorithm performing maximum-likelihood (ML) soft-decision syndrome decoding based on the LRB is presented. Contrarily to previous LRB-based ML syndrome decoding algorithms, this algorithm is more conveniently implementable for codes whose codimension is not small. New sufficient conditions for optimality are derived. These conditions exploit both the ordering associated with the LRB and the structure of the code considered. With respect to MRR-based sufficient conditions, they present the advantage of requiring no soft information and thus can be preprocessed and stored. Based on these conditions, low-complexity soft-decision syndrome decoding algorithms for particular classes of codes are proposed. Finally, a simple algorithm is analyzed. After the construction of the LRB, this algorithm computes the syndrome of smallest Hamming weight among o(K/sup i/) candidates, where K is the dimension of the code, for an order i of reprocessing. At practical bit-error rates, for codes of length N/spl les/128, this algorithm always outperforms any algebraic decoding algorithm capable of correcting up to t+1 errors with an order of reprocessing of at most 2, where t is the error-correcting capability of the code considered.
This dissertation investigates the complexity of minimum-distance decoding (MDD) algorithms for general binary linear block codes, and also investigates the use of such codes in constructing error correcting/detecting codes that are effective against both random and unidirectional errors. A new MDD algorithm is proposed, and its complexity is analyzed and compared to that of other such algorithms. The question of whether all good codes satisfy certain properties required for efficient implementation of the algorithm is also considered. The performance of decoding algorithms, with the information bit error rate as the criterion instead of the block error rate, is also studied. New decoding rules which can yield lower information bit error rates than the MDD rule are proposed and evaluated. A new class of binary codes is developed which corrects random errors and detects all unidirectional errors. These codes are obtained by appending a suffix to each codeword of a linear block code. To evaluate the efficiency of the new codes, a thorough comparison with the best previously obtained codes is made.
This dissertation presents a novel approach to soft decision decoding for binary linear block codes. The basic idea of this approach is to achieve a desired error performance progressively in a number of stages. For each decoding stage, the error performance is highly bounded and the decoding is terminated at the stage where either near-optimum error performance or a desired level of error performance is achieved. As a result, more flexibility in the trade-off between performance and decoding complexity is provided. The proposed decoding is based on the reordering of the received symbols according to their reliability measure. In the dissertation, we evaluate the statistics of the noise after ordering. Based on these statistics, we derive two monotonic properties which dictate the reprocessing strategy. Each codeword is decoded in two steps: (1) hard-decision decoding based on reliability information and (2) reprocessing of the hard-decision decoded codeword in successive stages until the desired performance is achieved. The reprocessing is based on the monotonic properties of the ordering and is carried out using a cost function. A new resource test highly related to the reprocessing strategy is introduced to reduce the number of computations for each reprocessing stage. For short codes of lengths N ≤ 32 or medium codes with 32 < N ≤ 64 with rate R ≥ 0.6, near-optimum bit error performance is achieved in two stages of reprocessing with at most a computation complexity of o(K2) constructed code-words, where K is the dimension of the code. For longer codes, three or more reprocessing stages are required to achieve near-optimum decoding. However, most of the coding gain is obtained within the first two reprocessing stages for error performances of practical interest. The proposed decoding algorithm applies to any binary linear code, does not require any data storage and is well suitable for parallel processing. Furthermore, the maximum number of computations required at each reprocessing stage is fixed, which prevents buffer overflow at low SNR. Further reductions of the number of computations based on probabilistic or decomposable properties of the code considered are also investigated. The generalization of the algorithm to other channels is discussed. Finally, the discrete time channel associated with the AWGN model after ordering is greatly simplified. This simplification allows an important reduction of the number of computations required to evaluate the theoretical error performance of any algorithm based on a partial or total ordering.