that it is a measure of the uniform distribution of a sequence in $[0,1)^s$. Section \[s:examples\] deals with special cases of the spectral test, like diaphonies, the spectral test for pseudorandom number generators, various figures of merit for integer lattice points, like they appear in the context of good lattice points and rank-1 lattice rules. Finally, we show that the extreme as well as the star discrepancy can be approximated arbitrarily close by special cases of the hybrid spectral test.

Adding digit vectors {#s:addition}
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Digital sequences on the $s$-torus are sequences that arise from operations with digit vectors in some given integer bases $b_i\ge 2$, $1\le i\le s$. For the sake of simplicity, we restrict the following discussion to the one-dimensional case.

As we have seen in Section \[s:introduction\], and as the block ciphers IDEA [@Lai91a] and AES [@Daemen02a] illustrate in cryptography, two types of addition of digit vectors are in use in applications, addition without carry and addition with carry. By the duality principle, the first addition leads to Walsh functions and the second type to $b$-adic functions as the proper tools for the analysis of such sequences.

Are these two types of function systems sufficient to study 