Derivative Descendants of Cyclic Codes and Derivative Decoding

This paper defines cyclic and minimal derivative descendants (DDs) of an extended cyclic code from the derivative of Mattson-Solomon polynomials. First, it demonstrates that the cyclic DDs are the same extended cyclic code. It allows us to efficiently decode extended cyclic codes based on their cyclic DDs. Second, since the minimal DDs are equivalent codes, it also allows us to perform soft-decision decoding based on the minimal DDs with permutations. Simulation result shows that our proposed derivative decoding can be close to the maximum likelihood decoding for certain extended cyclic codes, including some extended BCH codes.