Polynomial invariants related to error-correcting codes and hyperplane arrangements

Reference No.	2024a018
Type/Category	Grant for General Research-Workshop（Ⅱ）
Title of Research Project	Polynomial invariants related to error-correcting codes and hyperplane arrangements
Principal Investigator	Norihiro Nakashima（Nagoya Institute of Technology, Faculty of Engineering・Associate professor）
Research Period	June 27,2024. ～ June 27,2024.
Keyword(s) of Research Fields	Error Correcting Codes, Hyperplane Arrangements, Lattices, Graphs, Matroids, Weight Enumerator, Coboundary Polynomials, Tutte Polynomials, Characteristic quasi-Polynomials
Abstract for Research Report	This research focuses on the relationship between the theory of error-correcting codes, which automatically correct errors in transmission channel, and hyperplane arrangements, which are collections of affine subspaces of codimension 1 in a finite-dimensional vector space. The Hamming weight enumerators, which is used in the coding theory to evaluate error correction algorithms, are essentially equivalent to the coboundary polynomials and Tutte polynomials for hyperplane arrangements. In general, the computational complexity of the Hamming weight enumerator is in exponential time, but since the theory of hyperplane arrangements can be used to compute an explicit form of the coboundary polynomial theoretically , there is a question whether this transformation formula for the coboundary polynomial and the Hamming weight enumerator can be applied to the study of the theory of coding theory.In recent years, invariants such as various generalizations of Hamming weight enumerators, characteristic polynomials obtained as special values of Tutte polynomials, and characteristic quasi-polynomials, have also been actively studied. As a continuation of the General Research-Workshop "Relationship between error-correcting codes and hyperplane arrangements and their applications (2022a020)" held in 2022, the purpose of this workshop is to provide a deeper understanding of polynomial invariants and related studies on error correcting codes and hyperplane arrangements.To this end, we invite a wide range of researchers in both fields to present the latest research results in their respective fields and to discuss them in order to lead to joint research. We expect that the Hamming weight enumerator of a code determined from a concrete hyperplane arrangement will be computed, that the theory of computation of the harmonic weight polynomial and Jacobi polynomial of an error-correcting code will be established, and that the relationship between characteristic quasi-polynomials and error-correcting codes will be clarified.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Norihiro Nakashima（Nagoya Institute of Technology・Associate professor） Tsuyoshi Miezaki（Waseda University・Professor）