By Bouyukliev I., Fack V., Winne J.

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**Extra info for 2-(31,15,7), 2-(35,17,8) and 2-(36,15,6) designs with automorphisms of odd prime order, and their related Hadamard matrices and codes**

**Example text**

K (m)), where k = (u, w). The k-tuple λ(m) is induced by the given reduced word as follows. Suppose that m is the chain w = x0 ✄ x1 ✄ · · · ✄ xk = u. 4), x1 = x0 t1 = s1 s2 . . si . . sq , where the deleted generator si is uniquely determined. Let λ1 (m) = i. Now repeat the process. After f steps, we have reached xf , and after f deletions, we have obtained a uniquely determined reduced subword expression xf = sj1 sj2 . . sjq−f , 1 ≤ j1 < j2 < · · · < jq−f ≤ q. Again, xf +1 = xf tf +1 = sj1 sj2 .

Def Assume now that (ii) holds. For brevity, let M (i, j) = y[i, j] − x[i, j] for all i, j ∈ [n]. If M (i, j) = 0 for all i, j ∈ [n], then x = y. Let (a1 , b1 ) ∈ [n]2 be such that M (a1 , b1 ) > 0 and M (i, j) = 0 for all (i, j) ∈ [1, a1 ] × [b1 , n] \ {(a1 , b1 )}. Then, y(a1 ) = b1 and x(a1 ) < b1 . Now, let (a2 , b2 ) ∈ [n]2 be the bottom right corner of a maximal positive connected submatrix of M having (a1 , b1 ) as the upper left corner. 4) that a2 < n and b2 > 1. Because of maximality, there exist c ∈ [a1 , a2 ] and d ∈ [b2 , b1 ] such that M (c, b2 − 1) = 0 and M (a2 + 1, d) = 0.

Ai . . ak b1 . . bj . . bq or t xy = ⎩ a1 . . ai . . ak b1 . . bj . . bq for some i and j. In the ﬁrst case, we then have xt = t x = a1 . . ai . . ak < a1 . . ai . . ak = x, and in the second, xty = t xy = xb1 . . bj . . bq , and hence ty = b1 . . bj . . bq < b1 . . bj . . bq = y. Thus, in both cases, we reach a contradiction. 9) just says that W has a greatest element. ” In this section, we derive some of its basic properties. We also discuss automorphisms of Bruhat order. 1 (i) If W is ﬁnite, there exists an element w0 ∈ W such that w ≤ w0 for all w ∈ W .