A Library for Formalization of Linear Error-Correcting Codes

Abstract

Error-correcting codes add redundancy to transmitted data to ensure reliable communication over noisy channels. Since they form the foundations of digital communication, their correctness is a matter of concern. To enable trustful verification of linear error-correcting codes, we have been carrying out a systematic formalization in the Coq proof-assistant. This formalization includes the material that one can expect of a university class on the topic: the formalization of well-known codes (Hamming, Reed–Solomon, Bose–Chaudhuri–Hocquenghem) and also a glimpse at modern coding theory. We demonstrate the usefulness of our formalization by extracting a verified decoder for low-density parity-check codes based on the sum-product algorithm. To achieve this formalization, we needed to develop a number of libraries on top of Coq’s Mathematical Components. Special care was taken to make them as reusable as possible so as to help implementers and researchers dealing with error-correcting codes in the future.