Relationship between error-correcting codes and hyperplane arrangements and their applications

Reference No.	2022a020
Type/Category	Grant for General Research-Workshop（Ⅱ）
Title of Research Project	Relationship between error-correcting codes and hyperplane arrangements and their applications
Principal Investigator	Norihiro Nakashima（Nagoya Institute of Technology, Faculty of Engineering ・Associate professor）
Research Period	June 16,2022. ～ June 16,2022.
Keyword(s) of Research Fields	error correctiong code, hyperplane arrangement, Hamming weight enumerator, Whitney polynomial, Tutte polynomial, characteristic quasi polynomial
Abstract for Research Report	Hyperplane arrangements, which are collections of co-dimension 1 subspaces in a finite dimensional vector space, have been a research subject in many research fields. In particular, it is related to the theory of error-correcting codes, which is a theory for automatically correcting errors that occur in transmission channels such as CDs, DVDs, and QR codes. Specifically, it is known that the Hamming weight enumerator, which is involved in evaluating the correction capability of error correction algorithms in coding theory, and the Whitney polynomial (or coboundary polynomial, Tutte polynomial) of the hyperplane arrangement have a simple transformation formula and are essentially the same polynomial. The Whitney polynomial, on the other hand, is obtained by computing the characteristic polynomial for all restriction arrangements of flats in the intersection poset. Therefore, if the hyperplane is hereditarily free and the exponents of each restriction arrangements are known, then the Whitney polynomial can be computed by Terao's factorization theorem. Thus, there is an approach from the study of free arrangements to the evaluation of the error correction capability of a code. The purpose of this conference is to better understand the still unexplored relationship between hyperplane arrangements and codes, including the approach to error-correcting codes from free arrangements. For this purpose, we would like to invite not only researchers of hyperplane arrangements and codes, but also a wide range of researchers related to both fields to present and discuss the latest research results in their respective fields. We expect that the Hamming weight enumerator of the code determined from a specific hyperplane arrangement will be calculated, and that new relationships between the two fields will be discovered.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Norihiro Nakashima（Nagoya Institute of Technology・Associate Professor） Takuro Abe（Kyushu University・Professor）