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Implementation of Galois Field for Application in Wireless Communication Channels MATEC Web of Conferences 210, 03012 (2018) https://doi.org/10.1051/matecconf/201821003012 CSCC 2018 Implementation of Galois Field for Application in Wireless Communication Channels Dikshita Sarma1,*, Manash Pratim Sarma2, Kandarpa Kumar Sarma3 and Nikos E. Mastorakis⁴ 123 Department of Electronics and Communication Engineering, Gauhati University, Assam 4 Technical University of Sofia, Sofia, Kilment Ohridski 8, Bulgaria Abstract. This paper discusses the implementation of Galois Field based codes for application in wireless communication channel. It discusses the use of Galois Fields outlining the basic performance of a digital communication system in terms of BER curves. The work further discusses the performance of these codes in Gaussian and Rayleigh Fading Channels. Keywords. Coding, Bit Error Rate (BER), Additive White Gaussian Noise (AWGN), Rayleigh Fading Channel, Signal to noise Ratio (SNR), Hamming Code, Bose-Chaudhuri-Hocquenghem (BCH) code, Galois Field. 1 INTRODUCTION Section 4 documents the experimental Results. Finally, Section 5 concludes the paper. Codes are used for data compression, cryptography, . error-correction, and networking. Codes are studied for the purpose of designing efficient and reliable data transmission methods [16]. This typically involves the 2 WIRELESS CHANNEL MODEL AND removal of redundancy and the correction or detection of CHARACTERISTICS errors in the transmitted data. In information theory and Wireless channel is an unguided channel, where the coding theory with applications in computer science and signals not only contain the direct Line of Sight (LOS) telecommunication, error detection and correction or waves; but also a number of signals as a result of error control are techniques that enable reliable delivery diffraction, reflection and scattering. This propagation of digital data over unreliable communication channels. type is termed Multipath, and this results in degradation Many communication channels are subjected to channel of the performances of the channel. In the design of noise, and thus errors may be introduced during communication systems for transmitting information transmission from the source to a receiver. In digital through physical channels, it is convenient to construct communications systems, the ultimate function of the mathematical models that reflect the most important physical layer is to as quickly and as accurately as characteristics of the transmission medium [2]. schemes possible transfer data bits in the media. Data can be in both Gaussian Channel and Rayleigh Fading Channel transmitted over copper cable, optical fibre or free space. Two basic measures of physical performance levels are related to the speed at which data can be transmitted 2.1. Additive White Gaussian Noise Channel (Data Rate) and data integrity when they arrive at the (AWGN) destination. The primary measure of data integrity is called the Bit Error Ratio, or BER. The bit error ratio can AWGN is often used as a channel model in which the be considered as an approximate estimate of the bit error only impairment to communication is a linear addition of probability. This estimate is accurate for a long time wideband or white noise with a constant spectral density interval and a high number of bit errors [17]. The (expressed as watts per hertz of bandwidth) and a remaining part of this paper is systematized in the Gaussian distribution of amplitude. It does not account following way: Section 2 discusses the basic concepts of for fading, frequency selectivity, interference, non- Wireless Channel Model and its characteristics. Section linearity or dispersion. AWGN is commonly used to 3 briefly describes the Galois Field in Hamming Code simulate background noise of the channel under study. and Bose-Chaudhuri-Hocquenghem (BCH) code. The AWGN channel is represented by a series of outputs © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). MATEC Web of Conferences 210, 03012 (2018) https://doi.org/10.1051/matecconf/201821003012 CSCC 2018 Yi at discrete time event index i. Yi is the sum of the field extension L/K the field K already contains a input Xi and noise Zi, where Zi is independent and primitive nth root of unity, then it is a radical extension identically distributed and drawn from a zero-mean and the elements of L can then be expressed using the normal distribution with variance N (the noise). The Zi nth root of some element of K [19]. If all the factor are further assumed to not be correlated with the Xi. groups in its composition series are cyclic, the Galois Yi = Xi+Zi [1]. group is called solvable, and all of the elements of the corresponding field can be found by repeatedly taking roots, products, and sums of elements from the base 2.2 Multi Path Fading Channels (Rayleigh fading field. It is known that a Galois group modulo a prime is channels) isomorphic to a subgroup of the Galois group over the Multipath fading channel is usually modelled as rationals [3]. Rayleigh fading channels. Rayleigh fading models assume that the magnitude of a signal that has passed 3 GALOIS FIELD IN HAMMING CODE AND through a communications channel will vary randomly, BOSE - CHAUDHURI - HOCQUENGHEM (BCH) or fade, according to a Rayleigh distribution-the radial CODE IN WIRELESS COMMUNICATION component of the sum of two uncorrelated Gaussian random variables [1]. From the study of abstract algebra, it is known that the The channel characteristics are usually modelled as most flexible yet the basic system of rules is that of the h = α e^jφ , where α is the Rayleigh distributed field. In a field, the operation of addition, subtraction, magnitude and is the phase uniformly distributed in the multiplication and division is performed. To represent interval [π; -π] each character in a field, a bigger field is needed. In the The BER is an average quantity and not an binary numeral system or base-2 number system, each instantaneous quantity as expressed as BER =ꭍ Q(√α ² value are represented with 0 and 1 [25]. To convert a P/ϭ² )2αe^- α ²/2 dα. The BER is related to the SNR by decimal numeral system or base-10 number system into the expression BER = ½( √SNR=SNR + 2). At high binary system, it is needed to represent a decimal in SNR, the expression above reduces to BER = ½ SNR terms of sums of aꭍ2ⁿ .That is, if x is the said decimal [17]. number then we wish to have x =ꭍ ꭍ€Naꭍ2ⁿ. The coefficients an is then written in descending order of n and all leading zeros are then omitted. The final result 2.3 Galois Field becomes the binary representation of the decimal x. Galois fields, named after Evariste Galois, are used in Ultimately, binary system offers an alternative way of error-control coding, is an algebraic field with a finite representing the elements of a Galois Field. Both the number of members. A Galois field that has 2m polynomial and binary representation of an element have members is denoted by GF (2m), where m is an integer their own advantages and disadvantages. Since a bit is between 1 and 16. Galois theory helps us understand either 0 or 1, a bit is an element of gf (2). There is also a finite fields. Finite fields have numerous real-life byte which is equivalent to 8 bits thus is an element of gf applications in coding theory and combinatorial designs. (2⁸ ) [3]. As a result of applications in a wide variety of areas, There are several types of codes and the major finite fields are increasingly important in several areas of classification is the linear and the non-linear codes. mathematics, including linear and abstract algebra, Linear codes may be encoded using the method of linear number theory and algebraic geometry, as well as in algebra and polynomial arithmetic. The basic idea lies in computer science, statistics, information theory, and breaking information into chunks, appending redundant engineering [3]. Galois theory originated in the study of check bits to each block, these bits being used to detect symmetric functions the coefficients of a monic and correct errors. Each data and the check bits block is polynomial are the elementary symmetric polynomials in called the codeword. A code is linear when each the roots. Let K be a field. Given any polynomial f(X) K codeword is a linear combination of one or more [X] having distinct roots, the splitting field L of f(X) codewords. This concept is from linear algebra and often over K is a finite, normal and separable extension. The the codewords are referred to as vectors for that reason. essence of Galois theory lies in the association of a Again one of the characteristics of some block codes is group G, known as Galois group, to such a polynomial cyclic nature. That means any cyclic shift of codeword is or more generally, to an extension L/K with the above also a codeword and hence, linear, cyclic, block code properties. A main result of Galois Theory establishes a codewords can be added to each other and shifted one-to-one correspondence between the subgroups of G circularly in any way, and the result is a still a codeword. and the sub-fields of L containing K. The notion of a Since the sets of codewords may be considered as a solvable group in group theory allows one to determine vector space and also may be generated through whether a polynomial is solvable in radicals, depending polynomial division, therefore there are two methods of on whether its Galois group has the property of performing computation-linear algebra and polynomial solvability. In essence, each field extension L/K corresponds to a factor group in a composition series of arithmetic polynomial.
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