Number Theory, Coding Theory, and Their Applications in Machine Learning

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Talks

• Kim, Sung Whan (Hanbat National University)

Title: A New Sinogram Correction Method for Beam-Hardening Artifact Reduction

Abstract: In this work, we consider the challenging problem of reducing beam-hardening artifacts in CT images caused by materials with high density or high atomic number such as bones and artificial metal implants. Beam-hardening is an inevitable phenomenon in the polychromatic CT system.  We correct the beam hardened projection data measured by a CT machine using the evaluation of the mean energy of the projection data along each X-ray path. The proposed reduction algorithm in this paper is a new sinogram correction method for beam-hardening artifact and our numerical experiments validate that the proposed method effectively reduces beam-hardening artifacts. Furthermore, the proposed method is applicable to cases involving multiple and different metallic objects.

• Chang, Gyu-Whan (Incheon National University)

Title:  Multiplicative Ideal Theory in Commutative Algebra

Abstract: Multiplicative Ideal Theory is a branch of commutative ring theory that mainly studies the ideal structure and factorization properties of rings. In this talk, I will introduce the main research topics and results of this field focusing on what I've been working on.  

• Han, Sunghyu (KoreaTech)

Title: Square elements in Galois rings and MDS self-dual codes 

Abstract: Let GR(2^m, r) be a Galois ring with even characteristic. We prove that if r is even and n ≡ 0 (mod 4), then −(n−1) is a square element in GR(2^m, r) for all m ≥ 1. Using this fact we also prove that if (n−1) | (2^r − 1), 4 | n, and r is even, then there exists an MDS(Maximum Distance Separable) self-dual code over GR(2^m, r) with parameters [n, n/2, n/2 + 1]. 

• Cho, Jin-Hwan (NIMS)

Title: 팔레트 적재 문제에 대한 산업수학적 접근

Abstract: 팔레트 적재  문제는 산업공학에서 50년 이상 다루어 온  산업계의 유명한 문제이다. 클래식한  형태의 팔레트 적재 문제는 주어진  크기의 팔레트 위에 동일한 크기의 상자 들을 얼마나 많이  적재할 수 있는가를  찾는 최적화 문제로 이해할 수 있으며, 이때 상자는 90도 회전만을 허용한다.  팔레트 위에 적재된 상자들의 모양은 상자 크기에 대한 팔레트 크기의 효율적 분할에 의해 결정된다는 사실이 이미 밝혀졌다. 이 강연에서는 산업현장에서 나타나는 팔레트 적재 문제의 해결 사례와 함께 이러한 효율적 분할에 대한 조합론적 접근을 통해 산업수학적 문제 해결에 대해 설명한다.

• Park, Young Ho (Kangwon National University)

Title: Reflections of Journey Through Time and Topics

Abstract: I would like to take this time to share my reflections on my 30 years of service as a professor, both as an educator and a researcher.

• Ji, Jun Ho (Kangwon National University)

Title: Optimizing Persistence Images via Gradient Descent

Abstract: Persistent Homology (PH) is one of the most well-known tools in Topological Data Analysis, used to extract topological features from point cloud data by applying simplicial homology theory. The results of PH can be represented by persistence images, which can then be processed using statistical methods, including various machine learning tools. In this talk, we apply a gradient descent algorithm to optimize the weight and kernel parameters, which are key factors in generating persistence images. We demonstrate our approach by addressing a characterization problem in sewer data. This is joint work with Kyungbae Park.

• Jang, Deok-Kyu (Kangwon National University)

Title: Machine Learning for Mathematical Problems: Physics-Informed Machine Learning and Beyond

Abstract: The rapid advancement of machine learning has generated interest in its potential to address mathematical problems. In the field of partial differential equations (PDEs), two methods have emerged as particularly promising: Operator Learning, which seeks to model general solutions of PDEs using supervised learning, and Physics-Informed Neural Networks (PINNs), which employ neural networks as an ansatz for solving PDEs. This presentation will introduce these methods and discuss the research surrounding them. Additionally, it will examine efforts to integrate machine learning with various mathematical problems, highlighting some case studies.

• Nilay Kumar Mondal (Ewha Womans University)

Title: Binary few-weight codes using a mixed alphabet ring with applications in strongly walk-regular graphs 

Abstract: We use a mixed alphabet ring and suitable defining sets consisting of simplicial complexes to construct linear codes over a ring. We explicitly determine their Lee weight distributions and study their Gray images to obtain several families of few-weight binary linear codes. Most of the few-weight codes obtained are distance-optimal, self-orthogonal, and minimal. Moreover, we find an infinite family of binary three-weight projective codes, which produces strongly 3-walk-regular graphs as an application. This is a joint work with Yoonjin Lee. 

• Kim, Hyun Jin (Yonsei University)

Title: Construction of Binary Bisymmetric Self-dual Codes

Abstract: In this work, we introduce binary bisymmetric self-dual codes. We develop a construction method of binary bisymmetric self-dual codes by increasing its length from a small-length bisymmetric self-dual code. Using this method, we generate binary bisymmetric self-dual codes and find that many of these codes have favorable parameters. Also, we define the map from binary bisymmetric self-dual codes to reversible self-dual codes over the ring $\F_2+u\F_2$. This implies a one-to-one correspondence between the bisymmetric code over $\F_2$ and the reversible self-dual code over $\F_2+u\F_2$. Therefore, using this map on generated bisymmetric self-dual codes, we obtain reversible self-dual codes over $\F_2+u\F_2$, which were difficult to obtain using previously known methods. This work is jointed with Whan-Hyuk Choi.

• Jong Yoon Hyun (Konkuk University)

Title: Characterization of weakly regular p-ary bent functions of ℓ-form 

Abstract: In this talk, we present the essential properties of weakly regular p-ary bent functions of ℓ-form, where a p-ary function is from Fpm to Fp. We observe that most of studies on a weakly regular p-ary bent function f with f(0) = 0 of ℓ-form always assume the gcd condition: gcd(ℓ − 1,p − 1) = 1. We first show that whenever considering weakly regular p-ary bent functions f with f(0) = 0 of ℓ-form, we can drop the gcd-condition; using the gcd-condition, we also obtain a characterization of a weakly regular bent function of ℓ-form. Furthermore, we find an additional characterization for weakly regular bent functions of ℓ-form; we consider two cases m being even or odd. Let f be a weakly regular bent function of ℓ-form preserving the zero element; then in the case that m is odd, we show that f satisfies gcd(ℓ,p − 1) = 2. On the other hand, when m is even and f is also non-regular, we show that f satisfies gcd(ℓ,p − 1) = 2 as well. In addition, we present two explicit families of regular bent functions of ℓ-form in terms of the gcd-condition. This is joint work with Jungyun Lee and  Yoonjin Lee.

• Choi, Whan-Hyuk (Kangwon National University)

Title: Self-orthogonality matrix and Reed-Muller codes

Abstract: In this talk, we introduce the self-orthogonality check matrix SO_k, a particular matrix originating from the Simplex code. We also introduce a binary vector l(G) containing the number of columns in a matrix G. For a code C having G as a generator matrix, we can check the self-orthogonality of C by computing SO_k l(G)^T. We also show that the matrix SOk is obtained by puncturing a Reed-Muller code R(r, m). Consequently, we solve the problem of how many extra columns are needed to make the shortest SO embedding code from a given binary [n, k] code for any k ≥ 2. Finally, we give Algorithm 13 for constructing optimal self-orthogonal codes and obtain many new optimal self-orthogonal codes of dimensions 5 and 6.  This is joint work with Jon-Lark Kim.

Contacts

• KRIMS : krims@kangwon.ac.kr

• EIMS : ims@ewha.ac.kr

• Whan-Hyuk Choi : whchoi@kangwon.ac.kr