9.2 The Macdonald Groups $G(2,m)$

We performed the coset enumeration of the groups $G(n, m), n = 2, m = 2, \ldots, 11, 15$ in order to see whether they behave uniformly with respect to coset enumeration as implemented in MRC 1.2 or not. Using the ''level''-framework (see Section 3) seven strategies (NONE, P-ALL, P-G, P-R, I-ALL, I-R, I-R-P: see Section 4) and seven orderings (kbo-A, kbo-B, kbo-a, kbo-b, ll-BbAa, syl-l-BbAa, syl-r-BbAa: see Section 5 & 8) were combined and the results were compared. Overall, P-ALL, P-G and I-R-P perform almost equally worst. That is, why they were not further considered. The results are tabulated in Appendix D on page ff.

For $m = 2$, the results for the orderings do not differ much for each of the strategies selected. The strategy NONE is the best with 17 maximal and 17 totally defined cosets. The next one is I-R with 42/42 to 47/47 maximal/total defined cosets. Then comes I-ALL with 46/46 to 71/71 and finally P-R with 49/49 to 71/71, I-ALL being better than P-R for all orderings computed except ll-BbAa were they are equal. Remarkably, no combination of strategies and orderings defines more cosets totally than maximal.

For $m = 3$, NONE and I-R perform almost equally good with less than 161 cosets defined, I-R being better for all orderings but syllable-right. The two other strategies define between 300 and 650 cosets, except for I-ALL together with the syllable-right ordering which defines 171/192 cosets. This is still worse than for NONE and I-R. I-ALL performs better than P-R for the kbo-A, kbo-a, kbo-b, length-lexicographic and syllable-right ordering.

From this point onwards the performance of I-ALL depends very much on the ordering with kbo-A, kbo-a, and the length-lexicographical ordering being bad, and kbo-B and kbo-b being good while the behaviour of the syllable orderings is changing.

For $m = 4$, the picture is different. While P-R is worst, and I-R better than NONE except for the syllable-right ordering, I-ALL is best for both syllable orderings with respect to the total number of cosets defined while being close to the other two with respect to the maximal number of cosets defined. For kbo-B and kbo-b, I-ALL performs better than NONE.

For $m = 5$, the picture changes again. Now, NONE and I-R perform almost equally, this time NONE being better for most orderings, I-R only being better than NONE for kbo-b. For kbo-a, the length-lexicographical ordering and the syllable orderings they perform almost equally good. P-R is still worst except for the length-lexicographical ordering and the syllable-left ordering where I-ALL is worst. Remarkably, I-ALL is best for syl-r-BbAa.

For $m = 6$, we get some kind of chaos. For kbo-A, kbo-B, and ll-BbAa strategy NONE is best while for the others I-R is best. For the latter NONE performs considerably worse. I-ALL performs better than I-R for kbo-B and better than NONE for kbo-b, else worse than both. This time, P-R performs better than I-ALL for the length-lexicographical and the syllable orderings but still worse than the other strategies.

For $m = 7$, we get a similar picture, but there are the following differences. NONE performs better than I-R for kbo-a. Both perform worse for syllable-left which performed good for $m = 6$. For the syllable orderings NONE, I-ALL and P-R perform equally bad. I-ALL and P-R perform almost equally bad for kbo-A and kbo-a.

For $m = 8$, we have the following differences to $m = 7$. NONE performs better than I-ALL for kbo-b but still worse than I-R. P-R performs better than I-ALL for kbo-A and kbo-a but considerably worse for kbo-b.

For $m = 9, 10, 11$ no more changes are observed for the kbos and the length-lexicographical ordering while the syllable orderings are still slowly changing with all strategies being rather close together. Remarkably, for $m = 10, 11$ I-ALL together with syl-l-BbAa is worst with at least about two times as many cosets defined compared to all other combinations.

Thus for $m = 8, \ldots 11, 15$ we get the following picture. For kbo-A, kbo-a, and ll-BbAa we have that NONE is best, followed by I-R with 50% more cosets defined, then come P-R and finally I-ALL with about 400 % more cosets defined than NONE. The gaps are increasing with increasing $m$. For kbo-B we have that NONE is better than I-ALL with 50 % more cosets defined, followed by I-R with 75 % more cosets defined and finally P-R with 200 % more cosets defined. For kbo-b I-R is best with 25 % less cosets defined than NONE which defines about the same number of cosets for all kbo orderings. As for kbo-B I-ALL is next with about 50 % more cosets defined than NONE and finally we have I-R with 450 % more cosets defined. While the picture stays the same for all $m = 8, \ldots 11$, we have that for NONE the number of cosets defined does only increase between 1 to 15 cosets for increasing $m$. For the other strategies the number of cosets increases between 50 and 2000 for increasing $m$. Finally, there are the two syllable orderings. Here the fact just mentioned comes into play. While for syl-l-BbA I-R is best for $m = 8, \ldots 11$ it gets nearer to NONE with increasing $m$. For $m = 15$ NONE is best followed by I-R, P-R, and I-ALL while we had the order I-R, I-ALL, P-R, and finally NONE for $m = 8$. About the same with a different order holds for syl-r-BbAa.