2013/11/29 by Randriambololona, Hugues · 3 citations
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Coding theory and cryptography #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Polynomial and algebraic computation #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1312.0022
openalex publication_date 2013/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.