Basics

Enrico Paolini; Gianluigi Liva; Balázs Matuz; Marco Chiani

doi:10.1017/9781316443354.003

2 - Basics

Published online by Cambridge University Press: aN Invalid Date NaN

Balázs Matuz and

Enrico Paolini: Affiliation:
University of Bologna
Gianluigi Liva: Affiliation:
German Aerospace Center, Wessling
Balázs Matuz: Affiliation:
Huawei Munich Research Center
Marco Chiani: Affiliation:
University of Bologna

Book contents

Get access

Summary

Basic definitions and tools for error correction: In Chapter 2, we provide the basic elements of classical error correcting codes, how to perform operations in finite fields, the decision rules, the structure and properties of classical block codes, and finally a description of the Reed–Solomon codes which are particularly important for the erasure channel.

Keywords

finite fields communication channels linear block codes classical erasure codes Reed–Solomon codes

Information

Type: Chapter
Information: Coding for Erasure Channels , pp. 12 - 48

DOI: https://doi.org/10.1017/9781316443354.003 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2026

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Book purchase

Temporarily unavailable

References

Shannon, C., “A mathematical theory of communication,” Bell Syst. Tech. J., vol. 27, no. 3, pp. 379–423, 623–656, July 1948.10.1002/j.1538-7305.1948.tb01338.xCrossRef Google Scholar

Barg, A., “At the dawn of the theory of codes,” Math. Intelligencer, vol. 15, no. 1, pp. 20–27, 1993.10.1007/BF03025254CrossRef Google Scholar

Calderbank, A. R., “The art of signaling: Fifty years of coding theory,” IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2561–2595, 1998.10.1109/18.720549CrossRef Google Scholar

Hamming, R. W., “Error detecting and error correcting codes,” Bell Syst. Tech. J., vol. 29, no. 2, pp. 147–160, 1950.10.1002/j.1538-7305.1950.tb00463.xCrossRef Google Scholar

Golay, M. J., “Notes on digital coding,” Proc. IEEE, vol. 37, p. 657, 1949.Google Scholar

Muller, D. E., “Metric properties of Boolean algebra and their application to switching circuits, Univ. of Ill., Digital Computer Lab,” Internal Report, Technical Report, 1953.Google Scholar

Muller, D. E., “Application of boolean algebra to switching circuit design and to error detection,” Trans. I.R.E. Professional Group on Electronic Computers, vol. EC-3, no. 3, pp. 6–12, 1954.10.1109/IREPGELC.1954.6499441CrossRef Google Scholar

Reed, I., “A class of multiple-error-correcting codes and the decoding scheme,” Trans. IRE Professional Group on Information Theory, vol. 4, no. 4, pp. 38–49, 1954.10.1109/TIT.1954.1057465CrossRef Google Scholar

Hocquenghem, A., “Codes correcteurs d’erreurs,” Chiffers, vol. 2, pp. 147–156, 1959.Google Scholar

Bose, R. and Ray-Chaudhuri, D., “On a class of error correcting binary group codes,” Information and Control, vol. 3, no. 1, pp. 68–79, 1960.10.1016/S0019-9958(60)90287-4CrossRef Google Scholar

Reed, I. and Solomon, G., “Polynomial codes over certain finite fields,” J. Soc. Indust. Appl. Math., vol. 8, no. 2, pp. 300–304, June 1960.10.1137/0108018CrossRef Google Scholar

Chen, C., Peterson, W. W., and E. Weldon Jr, “Some results on quasi-cyclic codes,” Information and Control, vol. 15, no. 5, pp. 407–423, 1969.CrossRef Google Scholar

Berlekamp, E., Algebraic coding theory. New York, NY, USA: McGraw-Hill, 1968.Google Scholar

Peterson, W. and Weldon, E., Jr., Error-correcting codes. Cambridge, MA, USA: MIT Press, 1972.Google Scholar

MacWilliams, F. and Sloane, N., The theory of error-correcting codes. Amsterdam, the Netherlands: North Holland Mathematical Libray, 1977, vol. 16.Google Scholar

Wicker, S. B. and Bhargava, V. K., Reed-Solomon Codes and Their Applications. New York, NY, USA: Wiley-IEEE Press, 1994.Google Scholar

McEliece, R. J., The theory of information and coding. New York, NY, USA: Cambridge University Press, 2004.10.1017/CBO9780511819896CrossRef Google Scholar

Ryan, W. E., Lin, S., and Wilson, S. G., Channel Codes: Classical and Modern. Cambridge University Press, 2024.10.1017/9781009335928CrossRef Google Scholar

Roth, R. and Seroussi, G., “On generator matrices of MDS codes,” IEEE Trans. Inf. Theory, vol. 31, no. 6, pp. 826–830, 1985.10.1109/TIT.1985.1057113CrossRef Google Scholar

Seroussi, G., “A systolic Reed-Solomon encoder,” IEEE Trans. Inf. Theory, vol. 37, no. 4, pp. 1217–1220, 1991.10.1109/18.86977CrossRef Google Scholar

Bloemer, J., Kalfane, M., Karp, R., Karpinski, M., Luby, M., and Zuckerman, D., “An XOR-based erasure-resilient coding scheme,” International Computer Science Institute, Berkeley, CA, Technical Report TR-95.048, Technical Report, 1995.Google Scholar

Plank, J. and Xu, L., “Optimizing Cauchy Reed-Solomon codes for fault-tolerant network storage applications,” in Proc. Fifth IEEE Int. Symp. Network Computing and Applications (NCA’06), 2006, pp. 173–180.10.1109/NCA.2006.43CrossRef Google Scholar

Accessibility standard: WCAG 2.2 A

Why this information is here

This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.

Accessibility Information

The PDF of this book complies with version 2.2 of the Web Content Accessibility Guidelines (WCAG), offering more comprehensive accessibility measures for a broad range of users and meets the basic (A) level of WCAG compliance, addressing essential accessibility barriers.

Content Navigation

Table of contents navigation
Allows you to navigate directly to chapters, sections, or non‐text items through a linked table of contents, reducing the need for extensive scrolling.

Index navigation
Provides an interactive index, letting you go straight to where a term or subject appears in the text without manual searching.

Reading Order & Textual Equivalents

Single logical reading order
You will encounter all content (including footnotes, captions, etc.) in a clear, sequential flow, making it easier to follow with assistive tools like screen readers.

Short alternative textual descriptions
You get concise descriptions (for images, charts, or media clips), ensuring you do not miss crucial information when visual or audio elements are not accessible.