Multiple Error Correction in Redundant Residue Number Systems: A Modified Modular Projection Method with Maximum Likelihood Decoding

Error detection and correction codes based on redundant residue number systems are powerful tools to control and correct arithmetic processing and data transmission errors. Decoding the magnitude and location of a multiple error is a complex computational problem: it requires verifying a huge number of different possible combinations of erroneous residual digit positions in the error localization stage. This paper proposes a modified correcting method based on calculating the approximate weighted characteristics of modular projections. The new procedure for correcting errors and restoring numbers in a weighted number system involves the Chinese Remainder Theorem with fractions. This approach calculates the rank of each modular projection efficiently. The ranks are used to calculate the Hamming distances. The new method speeds up the procedure for correcting multiple errors and restoring numbers in weighted form by an average of 18% compared to state-of-the-art analogs.

Complex-Valued Phase Transmittance RBF Neural Networks for Massive MIMO-OFDM Receivers

Multi-input multi-output (MIMO) transmission schemes have become the techniques of choice for increasing spectral efficiency in bandwidth-congested areas. However, the design of cost-effective receivers for MIMO channels remains a challenging task. The maximum likelihood detector can achieve excellent performance—usually, the best performance—but its computational complexity is a limiting factor in practical implementation. In the present work, a novel MIMO scheme using a practically feasible decoding algorithm based on the phase transmittance radial basis function (PTRBF) neural network is proposed. For some practical scenarios, the proposed scheme achieves improved receiver performance with lower computational complexity relative to the maximum likelihood decoding, thus substantially increasing the applicability of the algorithm. Simulation results are presented for MIMO-OFDM under 5G wireless Rayleigh channels so that a fair performance comparison with other reference techniques can be established.

On Ordinary Words of Standard Reed–Solomon Codes over Finite Fields

Reed–Solomon codes are widely used to establish a reliable channel to transmit information in digital communication which has a strong error correction capability and a variety of efficient decoding algorithm. Usually we use the maximum likelihood decoding (MLD) algorithm in the decoding process of Reed–Solomon codes. MLD algorithm relies on determining the error distance of received word. Dür, Guruswami, Wan, Li, Hong, Wu, Yue and Zhu et al. got some results on the error distance. For the Reed–Solomon code [Formula: see text], the received word [Formula: see text] is called an ordinary word of [Formula: see text] if the error distance [Formula: see text] with [Formula: see text] being the Lagrange interpolation polynomial of [Formula: see text]. We introduce a new method of studying the ordinary words. In fact, we make use of the result obtained by Y.C. Xu and S.F. Hong on the decomposition of certain polynomials over the finite field to determine all the ordinary words of the standard Reed–Solomon codes over the finite field of [Formula: see text] elements. This completely answers an open problem raised by Li and Wan in [On the subset sum problem over finite fields, Finite Fields Appl. 14 (2008) 911–929].

Evaluation of the union bound for the decoding error probability using characteristic functions

Introduction: Since the exact value of a decoding error probability cannot usually be calculated, an upper bounding technique is used. The standard approach for obtaining the upper bound on the maximum likelihood decoding error probability is based on the use of the union bound and the Chernoff bound, as well as its modifications. For many situations, this approach is not accurate enough. Purpose: Development of a method for exact calculation of the union bound for a decoding error probability, for a wide class of codes and memoryless channels. Methods: Use of characteristic functions of logarithm of the likelihood ratio for an arbitrary pair of codewords, trellis representation of codes and numerical integration. Results: The resulting exact union bound on the decoding error probability is based on a combination of the use of characteristic functions and the product of trellis diagrams for the code, which allows to obtain the final expression in an integral form convenient for numerical integration. An important feature of the proposed procedure is that it allows one to accurately calculate the union bound using an approach based on the use of transfer (generating) functions. With this approach, the edge labels in the product of trellis diagrams for the code are replaced by their corresponding characteristic functions. The final expression allows, using the standard methods of numerical integration, to calculate the values of the union bound on the decoding error probability with the required accuracy. Practical relevance: The results presented in this article make it possible to significantly improve the accuracy of the bound of the error decoding probability, and thereby increase the efficiency of technical solutions in the design of specific coding schemes for a wide class of communication channels.

An Embedding Strategy Using Q-Ary Convolutional Codes for Large and Small Payloads

Matrix embedding (ME) code is a commonly used steganography technique, which uses linear block codes to improve embedding efficiency. However, its main disadvantage is the inability to perform maximum likelihood decoding due to the high complexity of decoding large ME codes. As such, it is difficult to improve the embedding efficiency. The proposed q-ary embedding code can provide excellent embedding efficiency and is suitable for various embedding rates (large and small payloads). This article discusses that by using perforation technology, a convolutional code with a high embedding rate can be easily converted into a convolutional code with a low embedding rate. By keeping the embedding rate of the (2, 1) convolutional code unchanged, convolutional codes with different embedding rates can be designed through puncturing.