Derivative Descendants and Ascendants of Binary Cyclic Codes, and Derivative Decoding

TL;DR

This paper introduces derivative descendants of extended cyclic codes, enabling soft-decision decoding that approaches maximum likelihood performance, with potential applications to BCH codes.

Contribution

It defines cyclic and minimal derivative descendants of extended cyclic codes and demonstrates their use in near-optimal soft-decision decoding.

Findings

Derivative decoding approaches maximum likelihood performance.

Cyclic DDs are the same extended cyclic code.

Minimal DDs are equivalent codes used for decoding.

Abstract

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.