Some lengths for which CPP interleavers have weaker minimum distances than QPP interleavers

Abstract

Abstract The objective of this paper is to find an upper bound on the minimum distance for turbo codes with true cubic permutation polynomial (CPP) interleavers for some particular interleaver lengths. The method we have used consists in proofing that, for the considered interleaver lengths, every true CPP has an inverse true CPP. Then we have proved that some interleaver patterns appear for every CPP. We address interleavers of lengths of the form 8p or 24p, with p a prime number so that 3 ∣ ( p − 1 ) , used in classical 1/3 rate turbo codes with recursive systematic convolutional component codes having generator matrix G = [ 1 , 15 / 13 ] , in octal form. We prove that 27 is an upper bound on the minimum distance for these types of lengths. We also derive the coefficients of the inverse true CPP for a true CPP of the considered lengths. It is known that for these interleaver lengths an upper bound on the minimum distance for quadratic PP (QPP) interleavers is equal to 36, a much higher value compared to 27. Thus, the importance of the result in the paper consists in that, for interleaver lengths mentioned above, CPP interleavers perform weaker compared to QPP interleavers. Consequently, to find better PP interleavers their degree has to be at least five.