Non-binary Codes Research Articles

A High Spectral Efficient Non-Binary TTCM-assisted G2 STBC-OFDM for 4G Systems

paper deals with the MIMO-OFDM technique that candidate with the fourth generation (4G) of the wireless communication systems, this technique can provide high data rate transmission without increasing transmit power and expanding bandwidth, also it can efficiently use space resources and has a bright future. This paper presents the channel coding assisted STBC-OFDM systems, and employs the Coded Modulation techniques (CM), since the signal bandwidth available for wireless communications is limited. Furthermore, a non-binary Turbo Trellis Coded Modulation (TTCM) decoder-based multidimensional MAP algorithm- assisted G2 STBC-OFDM is design and implemented. 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 simulation results show that the performance of the TTCM decoding algorithm outperforms the binary decoding methods and suitable to deal with error control coding of the STBC- OFDM schemes. Keywordscodes, TTCM, non-binary error correcting codes, Rings

An Algebraic Method for Decoding q-ary Codes via Submodules of Z^n

In this paper, by using a relation between binomial ideal and submodules of Z^n in , a submodule associated with the integer programming (IP) problem is defined. By computing the reduced Grobner basis (RGB) of the submodule, the decoding problem of non-binary q-ary codes is considered as an integer program problem. Decoding complexity is investigated and the effective factors in complexity are also determined. Furthermore, an example of the decoding method for a 3-ary code is provided.

Presenting the basic essence of limiting generalizations paradigm by algebra predicate structures

At the present stage of computer technology advancement, there are problems of using sequential algorithms and exclusively binary encoding. These problems require creating computational tools with a new design and using non-binary coding techniques. In this study, a method of formal representation of elementary tests, their domains and pattern systems in the language of predicate algebra was developed for the first time. Formalization of the recalculation rules between the domains of different levels of generality, using the mathematical tool of the algebra of predicates, was carried out. The developed mathematical models, specifying the rules for domain value recalculation, are represented as the corresponding AP structures. For a hardware implementation of the obtained models, the method of presenting algebra-predicate structures in the form of associative-logic converters was used. The obtained AP structures can be used for creating intelligent parallel-action systems, operating in real time.

On non-binary quantum repeated-root cyclic codes

In this paper, based on the Steane's enlargement construction, three classes of non-binary quantum codes are constructed from classical repeated-root cyclic codes of length 2ps over 𝔽q with odd characteristic p. The exact minimum distances of these quantum codes are determined. This construction yields a quantum MDS code with parameters [[2p, 2p - 4, 3]]p and two good quantum codes with parameters [[2p, 2p - 7, 4]]p and [[2p, 2p - 10, 5]]p.

Concurrent Error Detection of Binary and Nonbinary OLS Parallel Decoders

This paper presents a concurrent error detection (CED) scheme for orthogonal Latin square (OLS) parallel decoders. Different from a CED scheme found in the technical literature that protects only the syndrome generator, the proposed CED scheme protects the whole OLS decoder for single stuck-at faults. This paper presents the detailed design and analysis of the proposed CED scheme and shows that it is strongly fault secured for single stuck-at faults. Extensive simulation results are also provided; different figures of merit such as area, power dissipation, gate depth, and coverage are assessed. It is shown that the proposed decoder designs for ( n, k) t-bit error correcting OLS codes (k = 16 ...256; t = 2 ...5) have reasonable overhead; for example, the average area overhead of the proposed CED is 35.5 (23.6) % compared with an OLS decoder with no CED (i.e., the previously reported CED scheme). However, the most significant advantage of the proposed scheme is that it achieves 100% fault coverage for the whole CED circuit, thus providing a very efficient and fully fault-tolerant implementation. The proposed CED is applicable to both binary and nonbinary OLS codes; the CED for a nonbinary OLS decoder achieves comparable or better results than a binary OLS decoder. Moreover, simulation shows that the proposed CED scheme is better than double modular redundancy.

Analysis and Enumeration of Absorbing Sets for Non-Binary Graph-Based Codes

In this work, we first provide the definition of absorbing sets for linear channel codes over non-binary alphabets. In a graphical representation of a non-binary channel code, an absorbing set can be described by a collection of topological and edge labeling conditions. In the non-binary case, the equations relating neighboring variable and check nodes are over a non-binary field, and the edge weights are given by the non-zero elements of that non-binary field. As a consequence, it becomes more difficult for a given structure to satisfy the absorbing set constraints compared to the binary case. This observation in part explains the superior performance of non-binary codes over their binary counterparts. We show that the conditions in the non-binary absorbing set definition can be simplified in the case of non-binary elementary absorbing sets. Based on these simplified conditions, we provide design guidelines for finite-length non-binary codes free of small non-binary elementary absorbing sets. These guidelines demonstrate that even under the preserved topology, the performance of a non-binary graph-based code in the error floor region can be substantially improved by manipulating edge weights so as to avoid small absorbing sets. Our various simulation results suggest that the proposed non-binary absorbing set definition is useful for a range of code constructions and decoders. Finally, by using both insights from graph theory and combinatorial techniques, we establish the asymptotic distribution of non-binary elementary absorbing sets for regular code ensembles.

A Novel Decoding Method for Non-Binary TCM Codes

A new non-binary decoding method, which is called 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, decoding reliability for high states TCM codes that suitable for Fourth-Generation (4G), decreasing errors with increasing word length, and easy to implement with real-time applications. The proposed Yaletharatalhussein decoding algorithm deals with non-binary error control coding of the convolutional and TCM codes. Convolutional codes differ from block codes in that a block code takes a fixed message length and encodes it, whereas a convolutional code can encode a continuous stream of data, and a hard-decision decoding can easily be realized using the Yaletharatalhussein algorithm. 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 simulation results show that the performance of the nonbinary TCM-based Yaletharatalhussein algorithm outperforms the binary and non-binary decoding methods.

A Generalized Construction and Improvements on Nonbinary Codes From Goppa Codes

We present an efficient construction of extended length Goppa codes. Using this construction, we obtain 78 new nonbinary codes with better minimum distance than the previously known codes with the same length and dimension. The construction is based on the observation that certain Goppa codes can be seen as BCH codes.

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.

Self-orthogonal codes from some Bush-type Hadamard matrices

By means of a construction method outlined by Harada and Tonchev, we determine some non-binary self-orthogonal codes obtained from the row span of orbit matrices of Bush-type Hadamard matrices that admit a fixed-point-free and fixed-block-free automorphism of prime order. We show that the code [20, 15, 4]5 obtained from a (100, 45, 20) design is optimal, and those with parameters [36, 21, 6]3 and [20, 14, 4]5 obtained from a (36, 15, 6) and a (100, 45, 20) design respectively, are near-optimal for the given length and dimension. Furthermore, we obtained a conjecturally optimal self-dual doubly-even [72, 36, 12]2 code, and examined the code of an orbit matrix of a putative (676, 325, 156) design.

Models of outcomes for receiving information codes which do not contain zero symbols

Improvement of technological processes management information systems defines greater demands on the reliability of information systems, characterized by probabilities of outcomes receive information. In order to ensure high levels of reliability for long transmission there are widely used non-binary codes, which are actually pseudo-random sequences. Most preferred among them from the standpoint of immunity are codes not containing zero symbols. In this paper there are estimated the probabilities of outcomes of receiving information for the various models of communication channels with codes which do not contain zero symbols.

NB-JNCD Coding and Iterative Joint Decoding Scheme for a Reliable communication in Wireless sensor Networks with results

Privacy threat is a very serious issue in multi-hop wireless networks (MWNs) since open wireless channels are vulnerable to malicious attacks. A distributed random linear network coding approach for transmission and compression of information in general multisource multicast networks. Network nodes independently and randomly select linear mappings from inputs onto output links over some field. Network coding has the potential to thwart traffic analysis attacks since the coding/mixing operation is encouraged at intermediate nodes. However, the simple deployment of network coding cannot achieve the goal once enough packets are collected by the adversaries. This paper proposes non-binary joint network-channel coding for reliable communication in wireless networks. NB-JNCC seamlessly combines non-binary channel coding and random linear network coding, and uses an iterative two-tier coding scheme that we proposed to jointly exploit redundancy inside packets and across packets for error recovery.

Design of Binary and Nonbinary Codes from Lifting of Girth-8 Cycle Codes with Minimum Lengths

In this paper, some regular and irregular girth-8 cycle codes with arbitrarily high code rates are presented which have minimum lengths among all girth-8 cycle codes having parity-check matrices of the same row regularity. The parity-check matrices of the constructed girth-8 cycle codes can be considered as the mother matrices of some quasi-cyclic (QC) cycle codes with maximum-achievable girth 24 having better rates rather than the girth-24 Geometric and Cylinder-type cycle codes. Simulation results show that the binary and non-binary constructed cycle codes perform better than the repeat accumulate (RA) codes, progressive edge growth (PEG) codes and random-like LDPC codes of the same lengths and rates.

Grobner Bases for Lattices and an Algebraic Decoding Algorithm

In this paper we present Grobner bases for lattices given in a general form, including integer and non-integer lattices. Grdot{o}bner bases for binary linear codes were introduced by Borges-Quintana et al. . We extend their work to non-binary group block codes. Then, given a lattice Λ and its associated label code L, which is a group code, we define an ideal for L. A Grobner basis is assigned to Λ as the Grobner basis of its label code L. Since the associated label code for integer and non-integer lattices are group codes, the assigned Grobner bases can be obtained for both cases. Using this Grobner basis an algebraic decoding algorithm is introduced. We provide an example of the decoding method for a lower dimension lattice. We explain that the complexity of this decoding method depends on the division algorithm and show this decoding method has polynomial time complexity. Experiments for some versions of root lattices (E_7 and E_8) show that for low SNR the performance of these lattices is near to the lower bounds given in .

NONBINARY QUANTUM CODES DERIVED FROM REPEATED-ROOT CYCLIC CODES

Many good quantum error-correcting codes were constructed from cyclic codes. However, it is a difficult problem to determine the true minimum distance of quantum cyclic codes for large length n. In this work, we construct nonbinary quantum cyclic codes and asymmetric quantum cyclic codes that are derived from repeated-root cyclic codes for an arbitrary length ps, and determine the true minimum distance of all those codes. Some proposed quantum cyclic codes are optimal. Additionally, some proposed asymmetric quantum cyclic codes have better parameters than the ones available in the literature.

Non-Binary Coding for Texture Descriptors in Sub-Cellular and Stem Cell Image Classification

In recent years, binary coding of image features, such as local binary patterns and local phase quantization, have become popular in a large variety of image quantification tasks. Lately, some non-binary codings, such as local ternary pattern, have been proposed to improve the performance of these binary based approaches. In these methods it is very important to correctly choose the thresholds applied for building the coding used to represent a given image and its features by a feature vector. Keywords: Machine learning, non-binary coding, stem cell image, sub-cellular image, support vector machine, texture descriptors, Local Binary Pattern, Local Phase Quantization, Neighborhood Topology, Lysosome, Microtubules, Mitochondria, Mitochondria, Endoplasmic Reticulum, Plasma Membrane, Actin filaments

Hermitian dual containing BCH codes and Construction of new quantum codes

Let $q\geq 3$ be a prime power. Maximal designed distances of primitive Hermitian dual containing $q^{2}$-ary BCH codes (narrow-sense or non-narrow-sense) are determined by a careful analysis of properties of cyclotomic cosets. Non-narrow-sense BCH codes which achieve these maximal designed distances are presented, and a sequence of nested non-narrow-sense BCH codes that contain these BCH codes with maximal designed distances are constructed and their parameters are computed. Consequently, new nonbinary quantum BCH codes are derived from these non-narrow-sense BCH codes. The nonbinary quantum BCH codes presented here have better parameters than those quantum BCH codes available in the literature.

4-ary network coding for two nodes in cooperative wireless networks: exact outage probability and coverage expansion

The fact that sensor nodes are powered by limited-capacity batteries makes power efficiency; one of the most critical issues in wireless sensor networks (WSNs). Advanced communication techniques combined with network coding and cooperative schemes have attracted considerable attention as ways to improve power efficiency in wireless transmission as well as to achieve high throughput and spectral efficiency in WSNs. In this study, we consider cooperative wireless networks with two nodes and one base station, and investigate the effect of using non-binary network coding on the enhancement in power efficiency. First, we derive the exact and general outage probability in our network coding scheme. We show that full diversity order can be obtained using a non-binary network code with GF(4) in the considered network. We use this result to study the extent to which the coverage area of a wireless source node can be expanded by network coding without increasing transmit power. Our results indicate that the benefit in terms of coverage expansion is substantial. The results included in this study show the influence of optimal power allocation on power efficiency. The optimum ratio of power allocation varies according to the wireless channel environments and the field size of network codes.

Transmission scheme for a K -way relay multiple-input multiple-output channel

In the multiple-input multiple-output Gaussian K-users relay interference channel, each user intends to convey a multicast message to all the other users while receiving K−1 independent messages from the other users via an intermediate relay. An encoding and decoding strategy is proposed which is inspired from cryptography theory. The proposed encoding and decoding strategy involves a signal space alignment for an encryption message for multiple access phases and zero forcing with successive network code decoding for the broadcast phase. Then, the investigation of the antenna configuration conditions is an initial step for addressing the degrees of freedom of the above channel. It presents a new modified high-order pulse amplitude modulation that enables us to use the computationally efficient binary channel coding under a physical-layer network coding. It is shown that the proposed signalling scheme significantly outperforms the conventional non-binary code and is slightly worse than the lattice code.

Architecture of Generalized Bit-Flipping Decoding for High-Rate Non-binary LDPC Codes

A VLSI architecture for the generalized bit-flipping decoding algorithm for non-binary low-density parity-check codes is proposed in this paper. The tentative decoding steps of the algorithm have been modified to avoid computing and storing a matrix of dimension N×2 q , for a code (N,K) over GF(2 q ), reducing its complexity with a minimal penalization of its performance, less than 0.05 dB compared with the original algorithm. The architecture was synthesized using a 90 nm standard cell library, for the (837,723) non-binary code over GF(25), requiring 590220 xor gates and achieving a throughput of 89 Mbps. Additionally, it was implemented in a Virtex-VI FPGA device with a cost of 4070 slices and a throughput of 44.6 Mbps.