Corollary 4 Clause Samples

Corollary 4. For ∆ = √1 and every distribution PXM , Protocol 6 has a probability of error sn vanishing to 0 as n → ∞ and average length |π|av less than12 nRCO(M|PXM ) + O(√n log n). Furthermore, for a fixed R > 0, the fixed-length variant of Protocol 6 has probability of error sn vanishing to 0 as n → ∞ for all distributions PXM that satisfy R > RCO (M|PXM ) + O .√n−1 log nΣ .‌

Corollary 4. If D is a quaternion division algebra over an Ai(2)-field k and σ is of the first kind, then u+(D) ≤ 3 · 2i−2 and u−(D) ≤ 2i−2; = 3 · 2

Corollary 4. In the case G = D4, the subgroup of GL(g1) stabilizing both the quartic form and the skew-symmetric bilinear form, Stab(q, ⟨−, −⟩), is SL3 uS3 Proof. The previous theorem and the fact that SL3 and the diagram auto- morphism stabilize both forms yield the following containments: SL3 uS3 ⊆ Stab(q, ⟨−, −⟩) ⊆ Stab(q) = ⟨SL3, µ4⟩ u S3. Since −1 ∈ SL2, we also have −1 ∈ SL3. Thus SL3 uS3 is an index 2 subgroup 2 of ⟨SL3, µ4⟩ u S3. However, the coset containing i, a primitive fourth root of unity, is not in Stab(q, ⟨−, −⟩) since ⟨ix, iy⟩ = −⟨x, y⟩ for any x, y ∈ g1. Therefore Stab(q, ⟨−, −⟩) = SL3 uS3.

Corollary 4. If D is a quaternion division algebra over a global function field k, then u+(D) = 3, u−(D) = 1, and u0(D) = 2. Proof. By Chevalley-Warning theorem [Che35; War35], every finite field is a C1- field. By ▇▇▇▇-▇▇▇▇ theorem [Lan52], every global function field is a C2-field. Since every C2-field is an A2(2)-field [Lee13, between 2.1 and 2.2], by corollary 4.2.3, u+(D) ≤ 3 and u−(D) ≤ 1. By theorem 4.2.2, u0(D) ≤ 2. The equality follows from lemma 4.1.6 and lemma 4.1.1.