It is shown that not all Boolean functions can be realized by a two-level EXCLUSIVE-OR majority network. However, if repeats at the first level are allowed, then it is shown that such a network is universal. A minimal weight vector with respect to this latter network is defined. By using the restricted-affine-group (RAG) equivalence of Boolean functions, it is shown that if two functions are in the same RAG class, then they are realized by the same minimal weight vector to within permutations and/or sign changes.

It is shown that any decoding function for a linear binary code can be realized as a weighted majority of nonorthogonal parity checks. An example is given of a four-error-correcting code that is neither L-step orthogonalizable nor one-step majority decodable using non-orthogonal parity checks and yet is one-step weighted-majority decodable using only ten nonorthogonal parity checks.