Proposition

Definition 3 ([Tree code]). Fix integers s≥2 and m≥1 , and put k=(2s)^m, d=s^m. Let T be the complete rooted 2s -ary tree of depth m . The children of each internal node v are indexed by (g,b)∈[s]×0,1 . Its leaves are indexed by [k] . To …

Definition 3 ([Tree code]). Fix integers s≥2 and m≥1 , and put k=(2s)^m, d=s^m. Let T be the complete rooted 2s -ary tree of depth m . The children of each internal node v are indexed by (g,b)∈[s]×0,1 . Its leaves are indexed by [k] . To each internal node assign a table F_v:^s→-1,1 , and to each leaf assign a row R_i∈H . The word W(R,F)∈-1,1^A is defined pointwise as follows. At a∈ A , leaf i has sign (-1)^R_i(a) . An internal node with child signs (Z_g,b ) outputs (∏_g=1^s Z_g,0 )F_v(z), (-1)^z_g =Z_g,0 Z_g,1 . The root output is W(R,F)(a) .