  An interleaver is a critical component for the channel coding performance of
turbo codes. Algebraic constructions are of particular interest because they
admit analytical designs and simple, practical hardware implementation. Also,
the recently proposed quadratic permutation polynomial (QPP) based interleavers
by Sun and Takeshita (IEEE Trans. Inf. Theory, Jan. 2005) provide excellent
performance for short-to-medium block lengths, and have been selected for the
3GPP LTE standard. In this work, we derive some upper bounds on the best
achievable minimum distance dmin of QPP-based conventional binary turbo codes
(with tailbiting termination, or dual termination when the interleaver length N
is sufficiently large) that are tight for larger block sizes. In particular, we
show that the minimum distance is at most 2(2^{\nu +1}+9), independent of the
interleaver length, when the QPP has a QPP inverse, where {\nu} is the degree
of the primitive feedback and monic feedforward polynomials. However, allowing
the QPP to have a larger degree inverse may give strictly larger minimum
distances (and lower multiplicities). In particular, we provide several QPPs
with an inverse degree of at least three for some of the 3GPP LTE interleaver
lengths giving a dmin with the 3GPP LTE constituent encoders which is strictly
larger than 50. For instance, we have found a QPP for N=6016 which gives an
estimated dmin of 57. Furthermore, we provide the exact minimum distance and
the corresponding multiplicity for all 3GPP LTE turbo codes (with dual
termination) which shows that the best minimum distance is 51. Finally, we
compute the best achievable minimum distance with QPP interleavers for all 3GPP
LTE interleaver lengths N <= 4096, and compare the minimum distance with the
one we get when using the 3GPP LTE polynomials.
