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Extra resources for Artificial Evolution: European Conference, AE 95 Brest, France, September 4–6, 1995 Selected Papers
Zs−1 , 0, 0, . . ) and 0 ≤ zi ≤ b−1 for all i, then we have that sb (x) ≥ sb (z) with equality if and only if xi = zi i≡r (mod s) for all i = 0, . . , s − 1. (The lemma is simply the case where z = (b − 1, b − 1, . . , b − 1, 0, 0, . . ) (2) As a consequence of this lemma, we see that if (bs −1) | x and sb (x) = (b−1)s, then certainly (bt − 1) x for t > s. Using the notation introduced at the beginning of this section, we now have the following. 4. Let q = p s , p a prime, let m = de, where d > 2, e > 1 are integers.
Gm , say. Obviously, we may compute Ak by calculating the (i) contributions Ak from the component Gi to Ak , for i = 1, . . , m, and adding the results. As a consequence, in order to obtain the linear recurrence relation satisﬁed (i) by the numbers Ak , it is sufﬁcient to do this for the numbers Ak for each i. Since in general the size of the restriction of V1 to each of the components Gi is much smaller than V1 itself, this procedure greatly reduces the amount of computations. We applied the above method to compute minimal linear recurrence relations for various values of r for our counting problem deﬁned in (12) and (13).
These association schemes are studied by many authors. For instance: E. Bannai has kindly informed the authors of some hitherto unpublished work by N. Rao and D. K. Ray-Chaudhuri who studied these association schemes as the ‘group-case’ association schemes in early 80’s. ) I. Faradzev, M. Klin and M. Muzychuk  studied these schemes while they investigated S-rings. H. Sun  constructed these schemes as well as partially balanced incomplete block designs attached to them. The above description of these schemes are due to him.