Joint Permutor Analysis and Design for Multiple Turbo Codes

In this paper, we study the problem of joint permutor analysis and design for J-dimensional multiple turbo codes with J constituent encoders, J>2. The concept of summary distance is extended to multiple permutors of size N and used as the design metric. Using the sphere-packing concept, we prove that the minimum length-2 summary distance (spread) D_min,2 is asymptoticly upper-bounded by O(N ^J-1/J). We also show that the asymptotic minimum length-2L summary distance D_min,2L for the class of random permutors is lower-bounded by O(N^J-2J-epsi/), where epsi>0 can be arbitrarily small. Then, using the technique of expurgating "bad" symbols, we show that the spread of random permutors can achieve the optimum growth rate, i.e., O(N^J-1/J), and that the asymptotic growth rate of D_min,2L can also be improved. The minimum length-2 and length-4 summary distances are studied for an important practical class of permutors-linear permutors. We prove that there exist J-dimensional multiple linear permutors with optimal spread D_min,2 =O(N^J-1J/). Finally, we present several joint permutor construction algorithms applicable to multiple turbo codes of short and medium lengths