Further upper bounds on the minimum distance for turbo codes using CPP interleavers

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This paper addresses turbo codes with component convolutional codes as those in the Long Term Evolution (LTE) standard. We consider the interleaver lengths of the form L=32kLΨ, with kL∈{1,3} and Ψ is a product of prime numbers greater than three. For these interleaver lengths, we have shown that a true cubic permutation polynomial (CPP) f1x+f2x2+f3x3(modL), under the constraint f3=0(modpi) when a prime pi>3 and 3 does not divide (pi−1), always has an inverse true CPP, except when f3=L∕(4kL) and f2∈{L∕16,3L∕16,5L∕16,7L∕16}. For the previously mentioned exception, a true CPP has an inverse true quadratic PP (QPP). For CPP interleavers with true inverse CPPs, we have shown that the minimum distance is upper bounded by the values of 44, 36, and 27, for three different classes of coefficients. Previously, it was shown that for the same interleaver lengths and for QPP interleavers, the upper bound on the minimum distance is equal to 50. Due to the symmetry of the classical turbo codes, the minimum distance is also upper bounded by the value of 50 for CPP interleavers with a QPP inverse. The nonlinearity degrees for all considered CPP interleavers are computed. Several examples of dmin-optimal CPP interleavers and their inverse QPPs are given.