Binary coding by integration of polynomials

Based on the idea of integration and differentiation of polynomials, a large class of linear unequal error-protection (LUEP) codes is constructed. Many of these codes are optimal. A codeword is generated by binary discrete integration of an all-zero vector, using the information bits as integration constants. Decoding is performed by discrete differentiation of the received word. For special designs, all information bits are equally protected and in this case all classes of Reed-Muller codes are obtained. Thus, a new and very comprehensive description of these codes is given. The described codes are decoded by majority decisions over corresponding derivatives, based on the structure of Pascal´s triangle reduced modulo 2