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On products and powers of linear codes under componentwise multiplication

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arxiv 1312.0022 v3 pith:OED7Z37C submitted 2013-11-29 cs.IT math.AGmath.IT

On products and powers of linear codes under componentwise multiplication

classification cs.IT math.AGmath.IT
keywords symmetriccodescomponentwiselinearmultilinearmultiplicationpowersproducts
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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.

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