Identification of the Whole Interleaving Pattern From Randomly Interleaved Linear Block Codes With Known Minimum Distances

In this paper, an effective approach is proposed to recognize the order among codewords from the randomly binary and non-binary interleaved linear block coded streams if the code minimum distance is known, where the interleaver length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Nn</i> + <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> is not the multiple of the linear block code length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i>. Then, an efficient approach is developed without knowing linear block code parameters such as code rate and the code generator matrix (except the code length) to recognize the whole random interleaving pattern if the greatest common divisor (GCD) of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> is GCD(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n, d</i>) = 1 for both binary and non-binary interleaved linear block coded symbols. Further, for the case GCD(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n, d</i>) = <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> ̸= 1, the whole random interleaving pattern can be recognized with the probability of no more than (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i>!)<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−(<i>Nn</i>+<i>d</i>)/<i>q</i></sup> for the following cases: (1) the non-binary interleaved coded symbols without knowing the non-binary code generator matrix; (2) the binary interleaved coded symbols.