Application of Module to Coding Theory: A Systematic Literature Review

A systematic literature review is a research process that identifies, evaluates, and interprets all relevant study findings connected to specific research questions, topics, or phenomena of interest. In this work, a thorough review of the literature on the issue of the link between module structure and coding theory was done. A literature search yielded 470 articles from the Google Scholar, Dimensions, and Science Direct databases. After further article selection process, 14 articles were chosen to be studied in further depth. The items retrieved were from the previous ten years, from 2012 to 2022. The PRISMA analytical approach and bibliometric analysis were employed in this investigation. A more detailed description of the PRISMA technique and the significance of the bibliometric analysis is provided. The findings of this study are presented in the form of brief summaries of the 14 articles and research recommendations. At the end of the study, recommendations for future development of the code structure utilized in the articles that are further investigated are made

On products and powers of linear codes under componentwise multiplication

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.Comment: 75 pages; expanded version of a talk at AGCT-14 (Luminy), to appear in vol. 637 of Contemporary Math., AMS, Apr. 2015; v3: minor typos corrected in the final "open questions" sectio

Additive group actions on affine T-varieties of complexity one in arbitrary characteristic

Let X be a normal affine T-variety of complexity at most one over a perfect field k, where T stands for the split algebraic torus. Our main result is a classification of additive group actions on X that are normalized by the T-action. This generalizes the classification given by the second author in the particular case where k is algebraically closed and of characteristic zero. With the assumption that the characteristic of k is positive, we introduce the notion of rationally homogeneous locally finite iterative higher derivations which corresponds geometrically to additive group actions on affine T-varieties normalized up to a Frobenius map. As a preliminary result, we provide a complete description of these additive group actions in the toric situation.Comment: 31 page