Minimum distance upper bounds for 3-dimensional turbo codes using quadratic permutation polynomial interleavers

In this paper, we derive some useful upper bounds on the minimum distance d_min of a 3-dimensional (3-D) turbo code, first introduced by Berrou et al. (IEEE Information Theory Workshop, Lake Tahoe, CA, Sep. 2007) when using quadratic permutation polynomial (QPP) interleavers with a quadratic inverse. Furthermore, we give examples of interleaver lengths where an upper bound appears to be tight. Here, we consider binary (rate-1/2) upper and lower constituent encoders, while the original work of Berrou et al. consider double-binary (rate-2/3) constituent encoders. Finally, the results of a random search for good pairs of QPPs for use in the 3-D turbo code are presented, and the resulting 3-D turbo codes are compared to conventional turbo codes, both in terms of d_min and performance in the waterfall region. For instance, we have found a (6144, 2040) 3-D turbo code with an estimated d_min of 147.