Carrasco , Newcastle, UK has carried out research and given lectures in the area of channel coding, signal processing and mobile communication for over 20 years. He has published over papers, and much of his research relates to the design and construction of non-binary coding schemes for wireless communications, in particular Ring-TCM codes, Reed-Solomon codes and Algebraic-Geometric codes.

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First published: 28 November About this book Comprehensive introduction to non-binary error-correction coding techniques Non-Binary Error Control Coding for Wireless Communication and Data Storage explores non-binary coding schemes that have been developed to provide an alternative to the Reed — Solomon codes, which are expected to become unsuitable for use in future data storage and communication devices as the demand for higher data rates increases. Key Features: Comprehensive and self-contained reference to non-binary error control coding starting from binary codes and progressing up to the latest non-binary codes Explains the design and construction of good non-binary codes with descriptions of efficient non-binary decoding algorithms with applications for wireless communication and high-density data storage Discusses the application to specific cellular and wireless channels, and also magnetic storage channels that model the reading of data from the magnetic disc of a hard drive.

Author Bios Professor Rolando A. Free Access. Summary PDF Request permissions. Tools Get online access For authors. Matrix form of T l. We can note that h belongs to the kernel of R n , denoted as ker R n. In this case, a vector h is a parity check relation i. However, the opposite is not necessarily true.

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In GF q , the Hamming weight of a vector is the number of non-zero elements in this vector. The optimal threshold is determined by:. The normal distribution can be used to approximate the binomial probabilities of B l i when M is large:.

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So, in order to delimit the two behaviors of the variable B l i , it is necessary to compute the probability P i. In the case of binary codes, the probability P i has been calculated in [ 11 ]. But, it has never been studied in the general case of codes over GF 2 m. Our aim is to investigate this probability in the case of non-binary codes.

In the following, the theoretical study of P i is presented. This expression corresponds to that used in [ 11 ]. However, this threshold depends on the value of the error probability p e which is unknown for the receiver. So, the need to estimate this parameter is a blocking step in the almost dependent columns method and also leads to a lack of robustness. In order to address these problems, we propose a new iterative method based on the arithmetic mean of the variable B l i which do not depend on p e and where the iterative process permits to improve the detection probability.

We show here that the identification of the parameter n by our proposed method does not depend on the error probability p e. In this method, in order to improve the detection probability of n , an iteration process is introduced.

## Channel Coding Techniques for Wireless Communications | SpringerLink

We consider the idea of the iterative process proposed in [ 18 , 19 ]. These permutations permit to increase the probability to obtain non-erroneous pivots during the Gauss elimination. In this case, using the property 1, the mean E l will follow:.

Thereby, the mean E l is given by:. In the noiseless environment, the mean E l is stable at:. Function mode x : this operation provides the value which has the highest occurrence in the vector x. The proposed iterative method of the code word length identification is summarized in the Algorithm 2. The mean E l normalized by M , which is set to 1,, is represented in Figures 5 and 6. In Figure 5 , a zero probability of error i. According to 25 , the set is shown in Table 1.

The aim of our proposed algorithm is to blindly identify the length of non-binary code words in noisy environment. So, the average complexity is such that:. In order to analyze the performances of our blind identification method, the probability of correct detection of the code word length n is chosen as a performance criterion.

In the simulations, our method is applied to the non-binary LDPC codes which became candidate for future communication systems. For each simulation, 2, Monte Carlo trials are run where the data symbols are randomly chosen at each trial. In this part, we focus on:. Figure 7 shows the probability of detecting n according to p e for one, three, five, and ten iterations.

We can see that the gain between the first and the tenth iteration is significantly important.

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We can deduce that the iterative process improves significantly the detection performances of the blind identification method based on the mean calculation. In order to compensate and reduce the inter-symbol interference ISI caused by the multipath propagation, a linear mean square error MSE equalizer of length 20 was used. In Figure 8 , a comparison of the detection performances of our method in the case of AWGN channel is depicted. The gain between both is equal to 5 dB. A gain equal to 5 dB is exhibited. We have chosen to evaluate our proposed methods in the worst case of 8-PAM modulation because our aim was to show that our method has the best performances even in the case of the PAM modulation.

The detection probability of the method based on the mean calculation in the case of multipath Rayleigh channel. In the following, the performance study of the impact of n and q on the proposed method is presented. Figure 10 depicts the probability of detecting the correct n by our blind identification method according to the error probability p e in the cases of GF 4 , GF 8 , and GF We can deduce that the method based on the mean calculation is slightly sensitive to the increase of the Galois field dimension q.

To evaluate the detection performances of our blind identification method, the impact of increasing the code word length should be studied. Figure 11 shows the detection probabilities of n by the method based on the mean calculation. We can note that the increase of the code word length leads to lower detection performances with our proposed method. Impact of increasing n on the detection probability for LDPC codes. This probability can be improved by increasing the number of iteration of our algorithm. In this paper, we have introduced an algorithm devoted to the blind identification of the code word length for a non-binary code in a noisy transmission environment.

Using this algorithm, the code word length can be identified by calculating the arithmetic mean of the number of zeros that occur in the columns of the matrix obtained by the Gauss elimination. We have shown that the proposed algorithm is robust because it does not require the estimation of error probability, is insensitive to the high order of Galois field, and has the best detection performances for the most of modulation types.

Furthermore, this method provides better performances of detection when an iterative process is considered in order to increase the probability to obtain non-erroneous pivots during the Gauss elimination. Our future work will focus on identifying the remainder of the non-binary code parameters as well as a parity check matrix, permitting to implement a generic decoder in a noisy environment. Furthermore, a method based on using soft information that allows us to improve the performances of the blind identification algorithms will be published soon [ 31 ].

The two behaviors of B l i are limited in 7. The aim of this appendix is to demonstrate 8. Using these two probabilities, we will calculate the false alarm probability P fa , the probability of not detecting a theoretical dependent column P nd and the probability of detection P det. This probability can be determined by:. Calculation of the probability of detection P det : this probability is defined by:.

Using 32 and 34 , the optimal threshold can be determined by:. The probability P i is initially expressed by Henceforth, 15 becomes:.

Thereby, the probability P 1 s is determined by:. We demonstrate by the mathematical induction that the probability P 2 s can be expressed by:. The coefficients of this matrix correspond to the sum over GF 2 2 of the indexes of a row and a column. The computed probability verifies So, it can be written as:. Therefore, using 43 and 41 , the simplified expression of P 2 s is written as:. Using 37 and 44 , the overall probability P i is given by:. IEEE Commun. IEEE Trans.

J Stern, A method for finding code words of small weight. Coding Theory Appl. A Canteaut, F Chabaud, A new algorithm for finding minimum-weight words in a linear code: application to McElieces cryptosystem and to narrow-sense BCH codes of length A Valembois, Detection and recognition of a binary linear code.

Discrete Appl. MTA Review. XXII 4 , — Signal Process. Wireless Commun. I Reed, G Solomon, Polynomial codes over certain finite fields. Non-binary LDPC codes vs.

## Channel Coding Techniques for Wireless Communications

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