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Non-Binary Cyclic Codes and Its Applications in Decoding of High Dimensional Trellis-Coded Modulation

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Non-Binary Cyclic Codes and Its Applications in Decoding of High Dimensional Trellis-Coded Modulation NON-BINARY CYCLIC CODES AND ITS APPLICATIONS IN DECODING OF HIGH DIMENSIONAL TRELLIS-CODED MODULATION A Thesis Presented to The Faculty of the School of Electrical Engineering and Computer Science Fritz J. and Dolores H. Russ College of Engineering and Technology Ohio University In Partial Fulfillment of the Requirements for the Degree Master of Science by Biyun Zhou March, 2000 ACKNOWLEDGEMENT First, I would like to express my deepest thanks to my advisor, Dr. Jeffrey C. Dill for his guidance and instruction on this research project. His creative thinking and insightful directions made me follow a correct way in research. I also appreciated his effort to arrange a cooperative group meeting to provide an encouraging research atmosphere. I would also thank all the thesis committee members, Dr. Joseph Essman, Dr. Michael Braasch, and Dr. Sergio Lopez-Permouth for their helpful insights. I especially thank Dr. Sergio Lopez-Permouth for his instruction on the coding course, which guide me into this error control coding filed. His continues support on mathematics and patience in revising the initial version of this thesis is greatly appreciated. I would like to thank other group members in this research project, Xiaoxiao Cui, Sally Song and Yun yan for their help and discussions on the research. I thank Yung- Cheng Lo, former Ph.D. student who initiates this HDTCM project for his introduction and help when I first joined the group. I thank my family. Thanks to my father, a great telecommunications engineer for his guidance and encouragement in my academic advancement; and his always affection and support. I thank my mother for her sacrifice for the family and her love and anticipation. I thank my big brother and my sister-in-law for their tasty food each time I visit them. I especially thank my husband, Sen, who gave me love, support and encouragement all the time. TABLE OF CONTENTS ... ACKNOWLEDGEMENT ...................................................................................... ill TABLE OF CONTENTS ................... ........ ...................................................iv LIST OF FIGURES ............................ ..............................................................vi .. LIST OF TABLES ..................................................................................................vll CHAPTER 1 . INTRODUCTION ........................................................................................... 1 1.1 History of Error-Control Coding ..........................................................................1 1.2 Contribution of this Research ...............................................................................3 1.3 Outline of the Thesis ................. ......................................................................4 2 . BACKGROUND ......................... ... ............................................................ 5 2.1 Mathematical Background ..................................................................................5 2.1.1 Galois Fields ........................................................................................... 5 2.1.2 Polynomials over Galois Field .....................................................................7 2.1.3 Factoring xn - 1 .......................................................................................11 2.2 Communication System and Channel Coding ....................................................13 2.2.1 Coding and Modulation .............................................................................-14 2.2.2 Error Correction Codes ..............................................................................14 2.2.3 Trellis-Coded Modulation ..........................................................................15 2.3 High Dimensional Trellis-Coded Modulation .......... .. ................................ 18 3 . LINEAR BLOCK CODES. CYCLIC CODES AND BCH CODES ......... 20 3.1 Linear Block Code ..............................................................................................20 3.2 Cyclic Codes ..................................................................................................... 23 3.3 BCH codes ..........................................................................................................24 3.3.1 Encoding of BCH codes .............................................................................24 3.3.2 Erasures in BCH code .............................................................................. 28 3.4 Reed-Solomon Codes ..................................................................................... 29 3.5 Simulation Results ..............................................................................................31 4 . DECODING OF NON-BINARY BCH CODES ................. ..........................41 4.1 Overview of Decoding Algorithm for Non-binary BCH code ...........................41 4.2 Syndrome-Based Decoding and Berlekamp's Algorithm ..................................42 Standard Decoding Algorithm ...................................................................43 Berlekamp-Massey Algorithm ................................................................... 46 Erasure Decoding .......................................................................................48 Simulation Consideration ........................................................................... 50 4.3 Application of Non-Binary BCH Code in HDTCM ........................................... 53 5 . CONCLUSIONS ...........................................................................................57 5.1 Summary ............................................................................................................. 57 5.2 Future Research .................................................................................................. 59 References .............................................................................................................. -60 Appendix A List of some lengths Cary BCH codes ..........................................................62 B Generating generator polynomial for non-binary BCH codes .......................68 C Decoding of BCH Codes ................................................................................73 D Main Testing Function ...................................................................................78 LIST OF FIGURES Figure 2.1 : Block diagram for communication system ..................................................... 13 Figure 2.2. Rate % convolutional encoder with K=2 ........................................................ 15 Figure 2.3 Comparison between conventional coding/modulation and TCM ................... 17 Figure 3.1 : Decoding spheres ............................................................................................ 21 Figure 3.2. Sub-optimum BCH code selection algorithm ................................................. 28 Figure 3.3 Comparison of 4.ary. ~25.5primitive BCH codes and n=255 Reed-Solomon codes ...........................................................................................................................40 Figure 4.1 Standard concatenated coding system .............................................................. 55 Figure 4.2 Improved concatenated coding system ............................................................ 56 LIST OF TABLES Table 2.1 Construction of GF(8) .......................................................................................9 Table 2.2 Conjugacy class of GF(8) ................................................................................10 Table 2.3. Cyclotomic cosets modulo 3 1 with respect to GF(4) .....................................11 Table 2.4 Factorization of x2' .1 .................................................................................. 13 Table 3.1 Cyclotomic cosets and minimal polynomials for 4-ary BCH code of length 2 1 ....................................................................................................................................26 Table 3.2 List of 21 length 4-ary BCH codes ....................................................................31 Table 3.3 Acceptable 4-ary BCH code of lengths 3 1. 33. 35 and 39 ................................32 Table 3.5 Comparison of variable length 4-ary BCH codes ..............................................35 Table 3.6 cyclotomic cosets and minimal polynomials for primitive 4-ary BCH code ....37 Table 3.7 cyclotomic cosets modulo 65 with respect to GF(4) .........................................38 Table 3.8 list of 4-ary length 63 BCH codes .....................................................................39 CHAPTER 1 INTRODUCTION Research on error-control coding in the past fifty years has lead to the renovation of modem communications systems. We start this thesis by outlining a brief history of error- control coding in digital communication systems. We will introduce several technologies such as trellis-coded modulation, high dimensional trellis-coded modulation, and BCH codes. The primary contributions of this thesis research will also be explained in this chapter. We conclude the chapter with an outline of the thesis. 1.1 History of Error-Control Coding Concepts of error control coding can be traced back to Shannon's theorem, which states that channel capacity is related to bandwidth as well as the signal to noise ratio via the equation From this equation, one can infer that theoretically there exist perfect channels with infinite capacity.
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