Study On Low-power Encoders And Frequency-domain Decoding Based On FFT Of Reed-solomon Codes

In this dissertation, we mainly research for the novel encoding and decodingalgorithms.In the first part of this dissertation, low-power RS encoding algorithm and itsencoder has been proposed and implemented, by analyzing corresponding parameterssuch as generator polynomials and primitive polynomials etc. Multipliers play asignificant role in RS encoders, and therefore the appropriate choice of multipliers isable to ensure the low-power design of RS encoders without performance loss in errorcontrolling: the generator polynomial and primitive polynomial. On one hand, thegenerator polynomial of RS codes is defined as the polynomial which regards2tcontinuous elements in the finite field as its roots. However, the choice of the firstroot has not effect on the error correct performance. On the other hand, the choice ofprimitive polynomials only has the effect on algebraic architecture of finite fieldelements rather than error correct performance of decoding. This thesis proposes anefficient low-power RS encoding algorithm, in order to implement RS coders with theperfect combination of constant multipliers in finite field through selecting the mostproper generator polynomial and primitive polynomial.In the second part of this dissertation, algebraic soft-decision frequency-domaindecoding algorithm based on fast Fourier transform (FFT) in finite field has beenproposed. Because of the unique algebraic characteristics of finite field, we could takean efficient transformation in FT matrix, thus to meet the low-complexity computingrequirements. This thesis has mainly studies two kinds of FT methods. The firstmethod is called cyclotomic FFT. In this method, first of all we take the input vectorand the FT matrix into cyclotomic decomposes, which then are represented in normalbasis. After picking up common factors, original FT matrix has been decomposed intoseveral exactly same sub matrixes, and hence the single long FT computation hasbeen transformed into relatively shorter cyclic convolution computations. Anothermethod is called prime factors algorithm. First of all, the non-prime FT length hasbeen decomposed into several prime factors. By this mean, the original one-stage longFT could be decomposed into multi-stage relatively shorter FTs, thus to meet thelow-complexity computation requirement. Based on FFT in finite field, this thesis proposed a novel frequency-domain decoding algorithm. Firstly, we transformtime-domain codewords into frequency-domain codewords and then decode themwith KES just as exactly same as traditional Berlekamp-Massey in order to obtainerror location polynomials and error evaluation polynomials. In addition, the novelpolynomial selection significantly benefits to increase coding gain, especially whenthe length of the codeword is relatively short. As a result, the proposed novel algebrasoft-decision frequency-domain decoding algorithm improves the performance onboth coding gain and hardware requirements.