Algebraic Construction Schemes Of Low PMEPR Codes For The 4G OFDM Broad Band Wireless Communication

Orthogonal frequency division multiplexing(OFDM) technology is one of a multi-carrier modulation and demodulation techniques with many fine characteristics, andit is not only wildly used in many mature high-definition digital television(HDTV)transmission standards, but also considered as the international standard of the nextgeneration(4G) broadband wireless communication technology by IEEE together withmultiple-input and multiple-output(MIMO), space-time coding and smart antenna.Nevertheless, there is a big defect of OFDM, that is, a higher ratio of the peak energyof the OFDM's carriers to the mean energy, especially when the number of Sub-carriersis large, which will lead to many problems such as high power consumption of wirelessdevices, signal distortion, and other issues. PMEPR is one of the biggest obstacles forthe widely use of OFDM. Against to this defect, people from academia as well as in-dustry field have put forward a number of solutions. one of the most promising and es-sentially method to solve the problem is to algebraically construct low PMEPR codes.There are many breakthroughs in this field, for example, Rudin-Shapiro Polynomi-als(RSP) and Golay Complementary Sequences(GCS). Davis and Jedwab particularlyuse Boolean function to construct a sub set of Reed-Muller code with PMEPR≤2, i.e.,Davis-Jedwab Code(DJ-code), that is Golay Complementary Sequences(GCS). Thesecodes are identified as the coset of the first order Reed-Muller code inside the secondorder Reed-Muller code. Paterson, KG have further extended the Golay Complemen-tary Pairs to Golay Complementary Sets, and hence the code rate has a slight increase.However, the main drawback of Davis-Jedwab Code(DJ-code) is that its code rate isstill very small, especially when the length of the code is larger than 32. Therefore, how to construct the code with low PMEPR, large minimum distance as well as highcode rate for OFDM is still an open problem.The main purpose of this paper is to build some new algebraic constructionschemes of low PMEPR codes for the 4G OFDM broad band wireless communica-tion, mainly based on Boolean Functions, so to improve the code rate. The mainresults are listed as followings:1. We propose the definitions of Root Pairs and Sub-root Pairs. We have extendedthe Golay Complementary Pairs'power matching model, i.e., we have extendedthe model of energies equivalent to 2 to the model of energies less than or equalto a given arbitrary integral value k. Then we build a small and short lengthcode set, i.e., Sub-root Pairs as the initial root.2. We use the Boolean function theory and tensor matrix computation to buildsome long length Sub-root codes by the control of GCP and they have the samePMEPR as the short length Sub-root codes. As DJP is the special case ofGCP, we can put the boolean expression of DJP for simplification, then comesto a new method to build long length low PMEPR codes from short length lowPMEPR codes, and they have the same PMEPR. Furthermore, as these codes areconstructed as the subset of Reed-Muller codes, they have a good error correctioncapability, and also through simulation we have confirmed the improved coderate. We have extended the coset of the 1st order Reed-Muller code in secondorder Reed-Muller code to third, forth and even higher order Reed-Muller code.The core of this part is to build the main theorems and corollaries of constructinglong length and low PMEPR codes from the short ones.3. We extend the one-dimensional Root Pairs and Sub-root Pairs to the multi-dimensional cases, and by a similar construction method, we have constructed anew code set with a little bit higher PMEPR as well as an improved code rate.From this perspective, we have made a good trade-o? among code rate, errorcorrection capability and PMEPR.4. We propose the idea of partitioning Reed-Muller codes according to PMEPR,which provide an insight to fundamentally solve the problem of Coding for OFDMPMEPR.The research works of the construction schemes of low PMEPR codes for the 4thGeneration OFDM Broad Band Wireless Communication have far-reaching significance and important theoretical and commercial values. It will benefit much for the inde-pendent intellectual property rights for our country as well as for our high-definitiondigital television technology and broadband wireless communications technology's de-velopment.