We show how to use Simple Genetic Algorithm to produce Hadamard matrices of large orders, from teh full orthogonal design or oder 16 with 9 variables, OD(16; 1, 1, 2, 2, 2, 2, 2, 2, 2). The objective functionthat we use in our implementation of Simple Genetic Algorithm, comes from a Computational Algebra formalism of the full orthogonal design equations. In particular, we constructed Hadamard matrices of orders 144, 176, 208, 240, 272, 304 and 336, from the aforementioned orthogonal design. By varying three genetic operator parameters, we computer 62 inequivalent Hadamard matices of order 304 and 4 inequivalent Hadamard matrices of order 336. Therefore we established two new constructive lower bounds for the numbers of Hadamard matrices of order 304 and 336
