Actually, some of my constructions are flawed and/or not fully proven. The flaw is that when I used "constant weight codes" I used the minimum Hamming distance to make a simplex with all edge lengths exceeding some bound. But it was not necessarily the case that all edge lengths EQUALLED that bound (regularity). So to fix the flaw one would need to prove some sort of SHRINKAGE CONJECTURE that a simplex with all edge lengths >=K, can be "shrunk" to one with all edge lengths = K, contained in the old simplex. Is that true? No: a counterexample is a triangle with sides 1, 8, and 9.01. Another is sides 1, 1, 1.99. Another is the "flat sliver tetrahedron" with 4 sides=1 forming quadrilateral, other two sides=1.41. However it is true for well-enough-behaved triangles, e.g. if only one angle is less than 60. So recognizing this leads to the related question: what is the largest-volume regular simplex containable in a simplex? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)