Minimum free distance of CCSDS turbo encoders under (Truncated) Möbius interleavers

In this paper, we study the minimum free distance of the Consultative Committee for Space Data Systems (CCSDS) standard turbo encoder with Möbius interleavers. We introduce a new class of deterministic interleavers based on the Möbius interleavers called truncated Möbius interleavers and investigate their cycle structure. We find the exact number of total cycles of these interleavers and an upper bound on the binary fixed points in all of their cycles shifts. We derive the run time of the binary fixed point algorithm, which is used for estimating the minimum free distance of the employed turbo codes. Finally, we introduce some criteria to reduce the number of total cycles. This leads us to reduce the run time of the aforementioned algorithm.