We classify binary completely regular codes of length m with minimum distance δ for (m, δ) = (12, 6) and (11, 5). We prove that such codes are unique up to equivalence, and in particular, are equivalent to certain Hadamard codes. We prove that the automorphism groups of these Hadamard codes, modulo the kernel of a particular action, are isomorphic to certain Mathieu groups, from which we prove ...

Let X,Y be realcompact spaces or completely regular spaces consisting of Gδ-points. Let φ be a linear bijective map from C(X) (resp. C(X)) onto C(Y ) (resp. C(Y )). We show that if φ preserves nonvanishing functions, that is, f(x) 6= 0,∀x ∈ X, ⇐⇒ φ(f)(y) 6= 0,∀ y ∈ Y, then φ is a weighted composition operator φ(f) = φ(1) · f ◦ τ, arising from a homeomorphism τ : Y → X. This result is applied al...