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Asilomar Conference on Signals, Systems and Computers (ACSSC)

Fourier transform convolutional error-correcting codes

T.G. Marshall

Published Jan 1, 1989• 8 citations
Publisher
DOI
10.1109/acssc.1989.1200980

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We introduce the concept of convolutional factor graphs, which represent convolutional factorizations of multivariate functions, just as conventional (multiplicative) factor graphs represent multiplicative factorizations. Convolutional and multiplicative factor graphs arise as natural Fourier transform duals. In coding theory applications, algebraic duality of group codes is essentially an instance of Fourier transform duality. Convolutional factor graphs arise when a code is represented as a sum of subcodes, just as conventional multiplicative factor graphs arise when a code is represented as an intersection of supercodes. With auxiliary variables, convolutional factor graphs give rise to "syndrome realizations" of codes, just as multiplicative factor graphs with auxiliary variables give rise to "state realizations." We introduce normal and co-normal extensions of a multivariate function, which essentially allow a given function to be represented with either a multiplicative or a convolutional factorization, as is convenient. We use these function extensions to derive a number of duality relationships among the corresponding factor graphs, and use these relationships to obtain the duality properties of Forney graphs as a special case.
The minimum Hamming distance of a linear block code is the smallest number of bit changes required to transform one valid codeword into another. The code’s minimum distance determines the code’s error-correcting capabilities. Furthermore, The number of minimum weight codewords, a.k.a. error coefficient, gives a good comparative measure for the block error rate (BLER) of linear block codes with identical minimum distance, in particular at a high SNR regime under maximum likelihood (ML) decoding. A code with a smaller error coefficient would give a lower BLER. Unlike polar codes, a closed-form expression for the enumeration of the error coefficient of polarization-adjusted convolutional (PAC) codes is yet unknown. As PAC codes are convolutionally pre-transformed polar codes, we study the impact of pre-transformation on polar codes in terms of minimum Hamming distance and error coefficient by partitioning the codewords into cosets. We show that the minimum distance of PAC codes does not decrease; however, the pre-transformation may reduce the error coefficient depending on the choice of convolutional polynomial. We recognize the properties of the cosets where pre-transformation is ineffective in decreasing the error coefficient, giving a lower bound for the error coefficient. Then, we propose a low-complexity enumeration method that determines the number of minimum weight codewords of PAC codes relying on the error coefficient of polar codes. That is, given the error coefficient Awmin of polar codes, we determine the reduction X in the error coefficient due to convolutional pre-transformation in PAC coding and subtract it from the error coefficient of polar codes, Awmin - X. Furthermore, we numerically analyze the tightness of the lower bound and the impact of the choice of the convolutional polynomial on the error coefficient based on the sub-patterns in the polynomial’s coefficients. Eventually, we show how we can further reduce the error coefficient in the cosets.