Maximum spread of D-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 D-dimensional turbo code of frame size N. 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 root 2N. 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.