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On products and powers of linear codes under componentwise multiplication. (English) Zbl 1397.94119
Ballet, StĂ©phane (ed.) et al., Algorithmic arithmetic, geometry, and coding theory. 14th international conference on arithmetic, geometry, cryptography, and coding theory (AGCT), CIRM, Marseille, France, June 3–7, 2013. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-1461-0/pbk; 978-1-4704-2339-1/ebook). Contemporary Mathematics 637, 3-78 (2015).
Summary: In this text we develop the formalism of products and powers of linear codes under componentwise multiplication. As an expanded version of the author’s talk at AGCT-14, focus is put mostly on basic properties and descriptive statements that could otherwise probably not fit in a regular research paper. On the other hand, more advanced results and applications are only quickly mentioned with references to the literature. We also point out a few open problems.
Our presentation alternates between two points of view, which the theory intertwines in an essential way: that of combinatorial coding, and that of algebraic geometry.
In appendices that can be read independently, we investigate topics in multilinear algebra over finite fields, notably we establish a criterion for a symmetric multilinear map to admit a symmetric algorithm, or equivalently, for a symmetric tensor to decompose as a sum of elementary symmetric tensors.
For the entire collection see [Zbl 1317.11007].

MSC:
94B05 Linear codes, general
94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
14G50 Applications to coding theory and cryptography of arithmetic geometry
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