Formalization of Error-Correcting Codes: From Hamming to Modern Coding Theory
By adding redundancy to transmitted data, error-correcting codes (ECCs) make it possible to communicate reliably over noisy channels. Minimizing redundancy and (de)coding time has driven much research, culminating with Low-Density Parity-Check (LDPC) codes. At first sight, ECCs may be considered as a trustful piece of computer systems because classical results are well-understood. But ECCs are also performance-critical so that new hardware calls for new implementations whose testing is always an issue. Moreover, research about ECCs is still flourishing with papers of ever-growing complexity. In order to provide means for implementers to perform verification and for researchers to firmly assess recent advances, we have been developing a formalization of ECCs using the SSReflect extension of the Coq proof-assistant. We report on the formalization of linear ECCs, duly illustrated with a theory about the celebrated Hamming codes and the verification of the sum-product algorithm for decoding LDPC codes.
KeywordsFunction Node LDPC Code Computer Algebra System Variable Node Tanner Graph
T. Asai, T. Saikawa, K. Sakaguchi, and Y. Takahashi contributed to the formalization. The formalization of modern coding theory is a collaboration with M. Hagiwara, K. Kasai, S. Kuzuoka, and R. Obi. The authors are grateful to the anonymous reviewers for their comments. This work is partially supported by a JSPS Grant-in-Aid for Scientific Research (Project Number: 25289118).
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