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DFT matrix on GF(4)

  1. May 7, 2013 #1
    I am reading the following paper:
    Soft-decision decoding of polar codes with Reed-Solomon kernels

    On the last line of the page 319 (page 3 of the pdf) the author says "and [itex]G[/itex] is a Reed-Solomon kernel, which is in fact a DFT matrix".

    [itex]G[/itex] is defined on the page 321 (page 5 of the pdf) with possible change in the order of rows as
    G = \left( \begin{array}{cccc}
    1 & 1 & 1 & 0 \\
    1 & \alpha & \alpha^2 & 0 \\
    1 & \alpha^2 & \alpha & 0 \\
    1 & 1 & 1 & 1\end{array} \right),

    where [itex]\alpha[/itex] is a primitive element of [itex]\mathbf{F}_{2^2}[/itex].

    I do not understand why the author calls [itex]G[/itex] as a DFT matrix, because DFT matrix does not have zeros in its general form. The general form that I am considering is the following:
    Wikipedia Link.

    Can anyone explain the following:
    1. Why [itex]G[/itex] is a DFT matrix?
    2. If it is a DFT matrix, how can we implement it using FFT? I am looking for the butterfly structure that will implement it.

    Thanks for your time.
  2. jcsd
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