A partition of the hypercube into maximally nonparallel Hamming codes

By use of the Gold map, we construct a partition of the hypercube into cosets of Hamming codes that have minimal possible pairwise intersection cardinality. Let m ≥ 3 be odd and let F be the finite field GF(2) of order 2. Let σ be a power of 2, and assume that σ± 1 and 2− 1 are relatively prime (that is, both x → x and x → x are one-to-one mappings), which is, by simple arguments, equivalent to the condition gcd(s,m) = 1, where σ = 2. For example, σ = 2. We will treat the codes C of length 2 as collections of subsets of F , i.e., C ⊂ 2 . Recall some facts: (A) for all x, y ∈ F : (x+ y) = x + y (derived from (x+ y) = x+ y); (B) for all x ∈ F : x + x + 1 6= 0 (indeed, otherwise (x + 1) = (x + 1)(x + 1) = (x + 1)(x + 1) = x + x + x + 1 = x, which is impossible as f(x) = x is one-to-one); (C) the cardinality of the code B = {X ∈ 2 : ∑