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