Abstract
In this chapter we will discuss some applications of techniques from computational algebra and algebraic geometry to problems in coding theory. After a preliminary section on the arithmetic of finite fields, we will introduce some basic terminology for describing error-correcting codes. We will study two important classes of examples—linear codes and cyclic codes—where the set of codewords possesses additional algebraic structure, and we will use this structure to develop good encoding and decoding algorithms. Finally, we will introduce the Reed-Muller and geometric Goppa codes, where algebraic geometry is used in the construction of the code itself.
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© 1998 Springer Science+Business Media New York
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Cox, D., Little, J., O’Shea, D. (1998). Algebraic Coding Theory. In: Using Algebraic Geometry. Graduate Texts in Mathematics, vol 185. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-6911-1_9
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DOI: https://doi.org/10.1007/978-1-4757-6911-1_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98492-6
Online ISBN: 978-1-4757-6911-1
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