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Justesen code

    In coding theory, Justesen codes form a class of error-correcting codes which are derived from Reed-Solomon codes and have good error-control properties.

    Definition

    Let R be a Reed-Solomon code of length N = 2m-1, rank K and minimum weight N-K+1. The symbols of R are elements of F = GF(2m) and the codewords are obtained by taking every polynomial f over F of degree less than K and listing the values of f on the non-zero elements of F in some predetermined order. Let α be a primitive element of F. For a codeword a = (a1,...,aN) from R, let b be the vector of length 2N over F given by
    mathbf{b} = left( a_1, a_1, a_2, alpha^1 a_2, ldots, a_N, alpha^{N-1} a_N right)

    and let c be the vector of length 2N m obtained from b by expressing each element of F as a binary vector of length m. The Justesen code is the linear code containing all such c.

    Properties

    The parameters of this code are length 2m N, dimension m K and minimum distance at least
    sum_{i=1}^l i binom{2m}{i} .

    The Justesen codes are examples of concatenated codes.

    References