Towards Efficient Encoding Using Polynomials

We present an efficient new method of obtaining a generator matrix G from certain types of parity check matrices with a defective cyclic block structure. This novel approach describes parity check and generator matrices in terms of polynomials. Moreover, by using this polynomial algebra we have found efficient ways to implement the scheme. In addition, this method is as such interesting as it allows us to convert H into G without a systematic encoder in between (i.e., there is no diagonal subpart in the output). This alone is interesting as normally G would be dense if we were to form it from the given H by Gaussian elimination.