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In the previous five chapters, we have been primarily concerned with codes and coding techniques for channels on which transmission errors occur independently, i.e., random errors. However, there are communication and data-storage systems where errors tend to be localized in nature.
The objective of this chapter is to present some important elements of modern algebra and graphs that are pertinent to the understanding of the fundamental structural properties and constructions of some important classes of classical and modern error-correcting codes. The elements to be covered are groups, finite fields, polynomials, vector spaces, matrices, and graphs.
For 80 years, mathematics has driven fundamental innovation in computing and communications. This timely book provides a panorama of some recent ideas in mathematics and how they will drive continued innovation in computing, communications and AI in the coming years. It provides a unique insight into how the new techniques that are being developed can be used to provide theoretical foundations for technological progress, just as mathematics was used in earlier times by Turing, von Neumann, Shannon and others. Edited by leading researchers in the field, chapters cover the application of new mathematics in computer architecture, software verification, quantum computing, compressed sensing, networking, Bayesian inference, machine learning, reinforcement learning and many other areas.