Grassmannian codes from paired difference sets

On the value of the fifth maximal projection constant

“…Let X be the specific submatrix of Γ whose rows and columns are indexed by {x ∈ F 2s 2 : Q(x) = 1} and {x ∈ F 2s 2 : Q(x) = 0}, respectively. By Lemma 4.2 of [7], these two subsets of F 2s 2 are difference sets for F 2s 2 of cardinality k and l, respectively. As detailed in [7], this means that the rows and columns of X are equiangular, namely that (iii) and (iv) hold.…”

“…By Lemma 4.2 of [7], these two subsets of F 2s 2 are difference sets for F 2s 2 of cardinality k and l, respectively. As detailed in [7], this means that the rows and columns of X are equiangular, namely that (iii) and (iv) hold. Theorem 4.4 of [7] moreover gives that these two difference sets are paired, meaning that the columns of X form a tight frame for their span, so that (ii) holds.…”

“…. For any such s, k, l and m, the requisite matrix X of Proposition 3.1 is produced in the recent paper [7], albeit nonobviously so. In brief, let Q and B be the canonical hyperbolic-quadratic and symplectic forms on the binary vector space F 2s 2 , respectively: Let Γ be the corresponding character table of F 2s 2 , defined by Γ(x, y) = (−1) B(x,y) for all x, y ∈ F 2s 2 .…”

“…As detailed in [7], this means that the rows and columns of X are equiangular, namely that (iii) and (iv) hold. Theorem 4.4 of [7] moreover gives that these two difference sets are paired, meaning that the columns of X form a tight frame for their span, so that (ii) holds. Theorem 3.3 of [7] then implies that the rank of X is indeed m, so that (v) holds.…”