Dual forms of Reed-Muller expansions

The dual forms of Reed-Muller expansions based on the operations of logical equivalence and OR are investigated. The transforms describing the various fixed and mixed polarity product-of-sums expressions are derived and shown to be easily related to their counterparts for the normal sum-of-products forms. It is demonstrated that if the synthesis is restricted to using only the consistent fixed or mixed polarity Kronecker-Reed-Muller expansions, these dual forms can have lower weight than any normal form for some functions. It is also shown by employing extended function vectors, so that no restriction is placed on the form of solution, that the optimum weight dual and normal extended vectors differ by at most one term