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A Library for Formalization of Linear Error-Correcting Codes

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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.

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Notes

  1. The input and output alphabets are not the same for example in the case of the binary erasure channel that replaces some bits with an erasure.

  2. The MathComp library provides the same notation for sums, it is generic but cannot be used directly with the reals from the Coq standard library. One first needs to show that they satisfy the basic properties of appropriate algebraic structures and declare the latter as Canonical. We have chosen not to do that because it makes automatic tactics for reals such as field and lra inoperative. Our notation is therefore just a specialization of the generic notation provided by MathComp.

  3. This definition is specialized to communication over a noisy channel and is sufficient for our purpose in this paper. It can also be recast in a more general setting with conditional probabilities using an appropriate joint distribution [4].

  4. The evaluator polynomial is also denoted by \(\eta \) in the literature.

  5. Indeed, \(\sigma \) and \(\omega \) are coprime (see Sect. 5.1.2).

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Acknowledgements

T. Asai, N. Obata, K. Sakaguchi, and Y. Takahashi contributed to the formalization. The formalization of modern coding theory was a collaboration with M. Hagiwara, K. Kasai, S. Kuzuoka, and R. Obi. The authors are grateful to M. Hagiwara for his help. We acknowledge the support of the JSPS KAKENHI Grant Number 18H03204.

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Authors and Affiliations

  1. Cyber Physical Security Research Center, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Japan

    Reynald Affeldt

  2. Graduate School of Mathematics, Nagoya University, Nagoya, Japan

    Jacques Garrigue & Takafumi Saikawa

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  1. Reynald Affeldt
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  2. Jacques Garrigue
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Correspondence to Reynald Affeldt.

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Affeldt, R., Garrigue, J. & Saikawa, T. A Library for Formalization of Linear Error-Correcting Codes. J Autom Reasoning 64, 1123–1164 (2020). https://doi.org/10.1007/s10817-019-09538-8

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