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RA C1 tex Draft of 7 Jul 1999 11 47 a m RA Codes Achieve AWGN Channel Capacity Hui Jin and Robert J McEliece California Institute of Technology Pasadena California USA E mail hui rjm systems caltech edu Abstract In ref 3 we introduced a simplified ensemble of serially concatenated turbo like codes which we called repeat accumulate or RA codes These codes are very easy to decode using an iterative decoding algorithm derived from belief propagation on the appropriate Tanner graph yet their performance is scarcely inferior to that of fullfledged turbo codes In this paper we prove that on the AWGN channel RA codes have the potential for achieving channel capacity That is as the rate of the RA code approaches zero the average required bit Eb N0 for arbitrarily small error probability with maximum likelihood decoding approaches log 2 which is the Shannon limit In view of the extreme simplicity of RA codes this result is both surprising and suggestive 1 Introduction In ref 3 we introduced a class of turbo like codes the repeat and accumulate RA codes which are simple enough to allow a fairly complete theoretical analysis yet powerful enough to perform nearly as well as full fledged turbo codes The general idea is shown in Figure 1 An information block of length k is repeated q times scrambled by a pseudorandom permutation interleaver of size qk and then encoded by a rate 1 accumulator The accumulator can be viewed as a truncated rate 1 recursive convolutional encoder with transfer function 1 1 D but we prefer to think of it as a block encoder whose input block x1 xn and output block y1 yn are related by the formula y1 x1 y2 x1 x2 1 1 y3 x1 x2 x3 yn x1 x2 x3 xn In other words the outputs of the accumulator are the mod 2 partial sums of the inputs An RA code as just described is therefore a qk k linear block code but because of the unspecified interleaver there are a large number qk to be precise RA codes with these parameters For the theorist the most important thing about



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