To construct good DNA codes based on biologically motivated constraints, it is important that they have a large minimum Hamming distance and the number of GC-content is kept constant. Also, maximizing the number of codewords in a DNA code is required for given code length, minimum Hamming distance, and number of GC-content. In most previous works on the construction of DNA codes, quaternary constant weight codes were directly used because the alphabet of DNA strands is quaternary. In this paper, we propose new coding theoretic constructions of DNA codes based on the binary Hadamard matrix from a binary sequence with ideal autocorrelation. The proposed DNA codes have a greater number of codewords than or the equal number to existing DNA codes constructed from quaternary constant weight codes. In addition, it is numerically shown that for the case of codes with length 8 or 16, the number of codewords in the proposed DNA code sets is the largest with respect to the minimum reverse complementary Hamming distances, compared to all previously known results.