Non-binary Error-correcting Codes Research Articles

Blind identification of code word length for non-binary error-correcting codes in noisy transmission

Abstract In cognitive radio context, the parameters of coding schemes are unknown at the receiver. The design of an intelligent receiver is then essential to blindly identify these parameters from the received data. The blind identification of code word length has already been extensively studied in the case of binary error-correcting codes. Here, we are interested in non-binary codes where a noisy transmission environment is considered. To deal with the blind identification problem of code word length, we propose a technique based on the Gauss-Jordan elimination in GF(q) (Galois field), with q=2 m , where m is the number of bits per symbol. This proposed technique is based on the information provided by the arithmetic mean of the number of zeros in each column of these matrices. The robustness of our technique is studied for different code parameters and over different Galois fields.

Frequency and Quadrature-Amplitude Modulation for Downlink Cellular OFDMA Networks

The distribution of the intercell interference (ICI) in conventional cellular networks employing orthogonal frequency-division multiple-access (OFDMA) with quadrature-amplitude modulation (QAM) tends to approach a Gaussian distribution when all available subcarriers in each cell are fully loaded. Recently, it has been also shown that the worst-case distribution of the ICI as additive noise in wireless networks with respect to the channel capacity is Gaussian. Thus, the channel capacity in cellular networks is expected to be further enhanced when the ICI could be designed properly so that it has a non-Gaussian distribution. This observation motivates us to propose, in this paper, a downlink cellular OFDMA network employing a modulation scheme called frequency and QAM (FQAM). We also derive maximum-likelihood metrics for the binary or non-binary error-correcting codes employed in the proposed network and propose their practical sub-optimal versions. Numerical results demonstrate that the distribution of the ICI in the proposed network deviates far from the Gaussian distribution. As a result, the transmission rates for the cell-edge users in the proposed network are significantly improved. In addition, the measurement results using practically implemented FQAM-based OFDMA systems verify that the transmission rates for the cell-edge users can dramatically increase, compared with the conventional QAM-based OFDMA network.

A Non-Binary Trellis Code Modulation-based Novel Decoding Algorithm

The proposed system deals with Non-binary error control coding of the TCM scheme for transmissions over the AWGN channel. The idea of Non-binary codes has been extended for symbols defined over rings of integers, which outperform binary codes with only a small increase in decoding complexity. The basic mathematical concepts are necessary for working with Non-binary error-correcting codes are Groups, Rings and Fields. A new non-binary decoding method, Yaletharatalhussein decoding algorithm, is designed and implemented for decoding non-binary convolutional codes which is based on the trellis diagram representing the convolutional encoder. Yaletharatalhussein decoding algorithm outperforms the Viterbi algorithm and other algorithms in its simplicity, very small computational complexity, and easy to implement with real-time applications. The simulation results show that the performance of the non-binary TCM-based Yaletharatalhussein decoding algorithm outperforms the binary and non-binary decoding methods.

Construction of secure and fast hash functions using nonbinary error-correcting codes

This paper considers iterated hash functions. It proposes new constructions of fast and secure compression functions with nl-bit outputs for integers n>1 based on error-correcting codes and secure compression functions with l-bit outputs. This leads to simple and practical hash function constructions based on block ciphers such as the Data Encryption Standard (DES), where the key size is slightly smaller than the block size; IDEA, where the key size is twice the block size; Advanced Encryption Standard (AES), with a variable key size; and to MD4-like hash functions. Under reasonable assumptions about the underlying compression function and/or block cipher, it is proved that the new hash functions are collision resistant. More precisely, a lower bound is shown on the number of operations to find a collision as a function of the strength of the underlying compression function. Moreover, some new attacks are presented that essentially match the presented lower bounds. The constructions allow for a large degree of internal parallelism. The limits of this approach are studied in relation to bounds derived in coding theory.

Multi-Level Codes

The use of multi-level p.s.k. arouses interest in non-binary error-correcting codes. This paper points out that the algorithm for constructing BCH codes can be applied when the number of levels is a prime or a power of a prime. Quaternary codes are examined. Possible extensions to higher powers of two and to decimal codes are indicated.

Some properties of nonbinary error-correcting codes

An error-correcting code may be thought of as a subset S_0 of points belonging to a set S in which a metric is defined such that the distance between every pair of distinct points of S_0 is larger than some given number. In Hamming's original formulation, S was taken to be the set of all 2^n n -bit binary numbers and the distance between a pair of binary numbers s and t was taken to be the number of bits of s which do not agree with the corresponding bits of t . In this note we shall take S to be the set of all n -tuples in which each coordinate of an n -tuple can assume one of k integral values: 0, 1, \cdots, k - 1 , with k \geqq 2 . Properties of these nonbinary codes will be discussed.