Maximum Spread of -Dimensional Multiple Turbo Codes

This letter presents the mathematical framework involved in the determination of an upper bound of the maximum spread value of a -dimensional turbo code of frame size . This bound is named the sphere bound (SB). It is obtained using some simple properties of Euclidian space (sphere packing in a finite volume). The SB obtained for dimension 2 is equal to . This result has already been conjectured. For dimension 3, we prove that the SB cannot be reached, but can be closely approached (at least up to 95%). For dimensions 4–6, the construction of particular interleavers shows that the SB can be approached up to 80%. Moreover, from the SB calculation, an estimate of the minimum Hamming weight of the weight-two input sequence is derived.