High-order neural networks (HONN) are shown to decode some BCH codes in constant-time with very low hardware complexity. HONN is a direct extension of the linear perceptron: it uses a polynomial consisting of a set of product terms as its discriminant function. Because a product term is isomorphic to a parity function and a two-layer perceptron for the parity function has been shown by Rumelhart, Hinton, and Williams (1986), HONN has a simple realization if it is considered as having a set of parity networks in the first-half layer, followed by a linear perceptron in the second-half layer. The main problem in using high-order neural networks for a specific application is to decide a proper set of product terms. We apply genetic algorithms to this structure-adaptation problem.