Research

A matroid is a combinatorial structure that abstracts the notion of linear independence from linear algebra, and matroid theory provides a unified framework for properties shared by codes, graphs, finite geometry, and related objects introduced below.

I study matroid representation problems, with emphasis on representability by matrices over finite rings and the resulting structures.

Recent work includes descriptions of independence systems via modular independence over local rings, together with extension and comparison problems for representations.

I aim to clarify matroid structure through two-way interactions with independence notions arising in coding theory.

Coding theory studies the design of error-correcting codes and their algebraic/combinatorial properties.

I work mainly on linear codes over finite rings, focusing on critical problems, weight structures, and related characteristic polynomials.

Current topics include bounds on critical exponents for codes over finite chain rings and residue rings, and periodicity of weight enumerators.

Using the correspondence between codes and matroids, I seek quantitative combinatorial descriptions of code structure.

Finite geometry studies incidence structures on finite sets. Besides projective and affine geometries over finite fields, it also includes ring-based settings such as projective Hjelmslev geometry, chain geometry, and Laguerre geometry.

In my work, finite geometry (especially projective geometry) is primarily a tool rather than a standalone object of study.

Using points, subspaces, and incidence relations in projective spaces, I describe and organise the structures of codes and matroids in geometric terms.

From this viewpoint, I aim to clarify correspondences between algebraic objects and combinatorial structures, while providing insights useful for constructing optimal codes.

Enumerative combinatorics studies discrete structures through counting, generating functions, and polynomial invariants.

From an enumerative viewpoint, I study polynomial invariants arising from codes and matroids.

I analyse how weight enumerators, characteristic quasi-polynomials, and characteristic polynomials reflect algebraic conditions such as base rings and representation types.

Alongside theoretical bounds, I also work with concrete computations to capture detailed phenomena.