Abstract
This chapter deals with the problem of reliable transmission of digitally encoded information through an unreliable channel. In a way all channels are not completely reliable. Even the best telecommunication systems connecting numerous information centres in various countries have some non-zero error rate. Error-correcting codes considered in this chapter were designed to resolve this problem. After a giving an example of a non-linear code based on Hadamard matrices we switch to linear codes. We introduce generator and parity-check matrices and illustrate this technique with the classical Hamming codes. Then we consider polynomial codes and BCH-codes. We introduce non-binary codes and, most importantly, Reed-Solomon codes. In the last section we use non-binary codes to construct fingerprinting codes that give protection to intellectual property rights holders against colluding malicious users.
All sorts of computer errors are now turning up. You’d be surprised to know the number of doctors who claim they are treating pregnant men.
Isaac Asimov (1920–1992)
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Notes
- 1.
Richard Wesley Hamming (1915–1998) He participated in the Manhattan Project that produced the first atomic bombs during World War II. There he was responsible for running the IBM computers in Los Alamos laboratory which played a vital role in the project. Later he worked for Bell Labs after which he became increasingly interested in teaching and taught in a number of leading universities in the USA. Hamming is best known for his work on error-detecting and error-correcting codes. His fundamental paper on this topic “Error detecting and error correcting codes” appeared in April 1950 in the Bell System Technical Journal.
- 2.
Or watermarking, the war in terminology is currently raging.
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Staddon, J.N., Stinson, D.R., Wei, R.: Combinatorial properties of frameproof and traceability codes. IEEE Trans. Inf. Theory 47(3), 1042–1049 (2001)
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Slinko, A. (2015). Error-Correcting Codes. In: Algebra for Applications. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-21951-6_7
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DOI: https://doi.org/10.1007/978-3-319-21951-6_7
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