This page organises my work around representability over finite rings, q-polymatroids, and the connections among codes and finite geometry, focusing on the questions, the methods, and the directions that are currently active.
Research is the map of the project: the questions, methods, and current directions. For the representative papers and the full record, use Publications.
Start with These
This section highlights where the work is currently focused, why those directions matter, which papers best anchor them, and where collaboration points naturally arise.
Current focus
I work on matroid representability by matrices over finite rings, modular independence arising from local rings, and extension or comparison problems for representations.
Why this direction matters
It matters because it gives a coherent way to study independence and representability phenomena that do not appear over fields.
Natural overlap
This direction overlaps naturally with work on representability, local rings, and concrete matrix models.
Representative Paper
Koji Imamura, Keisuke Shiromoto - arXiv preprint arXiv:2603.08016, 2026
On Representing Matroids via Modular IndependenceCurrent focus
Using the critical problem for q-polymatroids as a guide, I explore how they connect with linear codes over finite rings, weight structures, and related combinatorial properties.
Why this direction matters
It matters because it puts the q-polymatroid formulation and quantitative coding-theoretic questions into a shared frame.
Natural overlap
It connects naturally with collaborations on finite-ring codes, weight structures, and critical problems.
Representative Paper
Koji Imamura, Keisuke Shiromoto - Discrete Math., 347(5), Paper No. 113924, 13, 2024
Critical problem for a q-analogue of polymatroidsCurrent focus
From finite-geometric and enumerative viewpoints, I aim to organise code-matroid correspondences and capture structures that can inform constructions of optimal codes.
Why this direction matters
It matters because the finite-geometric and combinatorial viewpoint can feed back into structural insight for constructing optimal codes.
Natural overlap
It also offers clear overlap with work on finite geometry, enumerative invariants, and constructions of optimal codes.
Representative Paper
Koji Imamura, Shinya Kawabuchi, Keisuke Shiromoto - arXiv preprint arXiv:2503.06830, 2025
On the one-dimensional extensions of q-matroidsHere the same three papers are placed as landmarks for the current direction, the q-polymatroid side, and the coding-theory side. The goal is to show where each result sits within the research questions and the present trajectory.
Plain-language summary
Some matroids become representable once fields are replaced by finite chain rings. This paper explains how to recognise that situation and how it connects to codes.
Technical summary
Using modular independence over local commutative rings, it gives criteria for representability over finite chain rings and describes the correspondence with codes.
Why start here
It is the clearest current entry point to the finite-ring representability side of the project and its links with codes.
Representative Paper
Koji Imamura, Keisuke Shiromoto - arXiv preprint arXiv:2603.08016, 2026
On Representing Matroids via Modular IndependencePlain-language summary
This paper identifies the critical problem to ask for q-polymatroids and supplies basic examples that make the framework usable.
Technical summary
It formulates the critical problem for q-polymatroids and introduces q-analogues of minimal blocks together with explicit examples.
Why start here
It provides the basic vocabulary and a concrete starting point for the q-polymatroid side of the work.
Representative Paper
Koji Imamura, Keisuke Shiromoto - Discrete Math., 347(5), Paper No. 113924, 13, 2024
Critical problem for a q-analogue of polymatroidsPlain-language summary
This paper gives upper bounds on how large the critical exponent of a code over a finite chain ring can be.
Technical summary
It studies the critical problem for codes over finite chain rings and proves upper bounds for the critical exponent.
Why start here
On the coding-theory side, it most clearly shows how the project leads to concrete quantitative results.
Representative Paper
Koji Imamura, Keisuke Shiromoto - Finite Fields Appl., 76, Paper No. 101900, 14, 2021
Critical Problem for codes over finite chain ringsThe three blocks below summarise the question, the approach, and why the topic matters.
When the base algebra is changed from a field to a finite ring, one encounters matroids that are not representable over fields, together with independence notions arising from local rings. I ask how these finite-ring phenomena appear in matroid representations.
I study representation problems over finite rings together with q-polymatroids, finite geometry, and code correspondences, in order to compare representations, study extensions, and isolate structures related to optimal codes.
This viewpoint helps explain discrete structures that are harder to see over fields, and provides combinatorial language for understanding known optimal codes and organising candidates for new constructions.
If you are new to my work, these two papers are the clearest way to grasp the current main themes and a sensible reading order.
Why Read This First
A clear entry point to the current main direction: the representation problem over finite rings and its correspondence with codes.
Recommended for
Recommended for readers who want the clearest view of the current main direction: representability over finite rings and its correspondence with codes.
On Representing Matroids via Modular Independence
Koji Imamura, Keisuke Shiromoto - arXiv preprint arXiv:2603.08016, 2026
At a Glance
Preprint studying matroid representations via modular independence over local commutative rings, giving criteria for representability over finite chain rings and connections with codes.
Why Read This First
A good entry point to the critical problem for q-polymatroids, with the formulation and representative examples together.
Recommended for
Recommended for readers entering from the q-polymatroid side and wanting the formulation of the critical problem together with basic examples.
Critical problem for a q-analogue of polymatroids
Koji Imamura, Keisuke Shiromoto - Discrete Math., 347(5), Paper No. 113924, 13, 2024
At a Glance
Formulates the critical problem for a q-analogue of polymatroids, and gives q-analogues of minimal blocks together with concrete examples.
Taken together, the recent papers show a trajectory from the critical problem for codes over finite chain rings, through q-polymatroids, to representation problems via modular independence over local rings.
This stage studies the critical problem for codes over finite chain rings and gives upper bounds for critical exponents, providing a concrete quantitative result on the coding-theory side.
Representative Paper
Koji Imamura, Keisuke Shiromoto - Finite Fields Appl., 76, Paper No. 101900, 14, 2021
Critical Problem for codes over finite chain ringsThe critical-problem viewpoint is extended to q-polymatroids by giving a formulation and basic examples, creating a bridge toward finite-geometry and q-analogue directions.
Representative Paper
Koji Imamura, Keisuke Shiromoto - Discrete Math., 347(5), Paper No. 113924, 13, 2024
Critical problem for a q-analogue of polymatroidsThe recent stage studies matroid representations via modular independence over local commutative rings, bringing together criteria for representability over finite chain rings and links with codes.
Representative Paper
Koji Imamura, Keisuke Shiromoto - arXiv preprint arXiv:2603.08016, 2026
On Representing Matroids via Modular IndependenceRecent talks have continued to focus on modular independence, representation problems over finite rings, and related coding-theoretic themes. This section is a quick way to see how the work is currently being presented.
IMI Cryptography Seminar - January 7 (Wed)
Why this talk matters
A recent seminar talk that presents the representation problem via modular independence over local rings in a concentrated form.
47th Australasian Combinatorics Conference (47ACC) - December 4 (Thu)
Why this talk matters
An international conference talk showing how independence systems based on modular independence are being presented to combinatorics audiences.
Designs, Codes, and Related Combinatorial Structures 2025 - December 14 (Sun)
Why this talk matters
A workshop talk showing how matroid representations via modular independence are being discussed in a codes-and-designs setting.
These topics mark the places where the current work most naturally overlaps with nearby research. If your work is close to them, this section is a quick way to see where questions or methods can meet.
This is a natural overlap point if you work on representability over finite rings, local rings, or comparisons among concrete matrix models, since the underlying questions and techniques are often shared directly.
Work on finite-ring codes, weight structures, and critical problems connects naturally here through q-polymatroids and the finite-chain-ring side of the project.
There is also clear overlap with work on finite geometry, enumerative invariants, and constructions of optimal codes through code-matroid correspondences and projective-geometric organisation.
The sections below add a little more detail on the themes behind the overview, while making their roles and objects of study easier to distinguish.
Keywords
Matroid theory studies structures that abstract linear independence and let us compare phenomena shared by codes, graphs, and finite geometry.
My central question here is how far matroids can be represented by matrices over finite rings, and what kinds of independence systems arise from modular independence over local rings.
I am interested in representability phenomena that disappear over fields, and in comparing them with independence notions coming from coding theory.
Keywords
In coding theory, I work mainly on weight structures and critical problems for linear codes over finite rings.
Current questions include bounds on critical exponents for codes over finite chain rings and residue rings, periodicity of weight enumerators, and their relation to characteristic polynomials.
Through correspondences with q-polymatroids and matroids, I aim for quantitative combinatorial descriptions of code behaviour.
Keywords
In my work, finite geometry is primarily a geometric tool for organising the structures of codes and matroids rather than a standalone object of study.
Using points, subspaces, and incidence relations in projective spaces, I arrange the structures of codes, q-polymatroids, and matroids, while keeping ring-based geometries in view as well.
The goal is to turn geometric organisation into clearer correspondences and insight for constructions of optimal codes.
Keywords
Enumerative combinatorics reads the structure of codes and matroids through counting and polynomial invariants.
I focus on how weight enumerators, characteristic quasi-polynomials, and characteristic polynomials reflect changes in base rings and representation types.
Alongside theoretical bounds, I use concrete computations to track where genuinely new phenomena appear.