arXiv subjects

Jon-Lark Kim

Publications and source records attributed to Jon-Lark Kim.

At least 19 recordsLinked to original sources

arXiv2026

New binary optimal LCD codes using heuristic embedding

In this paper, we investigate the construction of binary optimal LCD codes through short LCD embeddings. For this purpose, we design heuristic frameworks based on a greedy algorithm. We explore the search spaces of LCD embeddings using the fact that an invertible matrix together with an arbitrary matrix yields an LCD embedding. We therefore use elementary row operations on the invertible block and single entry-flips on the arbitrary block as local moves in a greedy algorithm. Using this method, we have found $14$ optimal new LCD codes with dimensions 7 and 8 for lengths from 55 to 201.

cs.IT↗

arXiv2026

Constraint-Preserving Genetic Algorithms for Embedding Linear Codes into Self-Orthogonal Codes

In this paper, we aim to construct binary optimal self-orthogonal codes using shortest self-orthogonal embedding methods. For this purpose, we design a heuristic framework based on a genetic algorithm. We explore the search space of shortest self-orthogonal embeddings using a fitness function based on the minimum distance and the number of minimum-weight codewords. We construct \emph{constraint-preserving} crossover and mutation operations so that every chromosome yields a valid self-orthogonal embedding, while high-fitness structural features, such as favorable subsequences of orthogonal generators, are propagated across generations. We also analyze the time and storage complexity of the algorithm, and validate our design through an ablation study on guided crossover and a comparison with random search under an equal time budget. Using this method, we obtain $66$ new binary optimal self-orthogonal codes that meet the upper bound, together with $135$ further self-orthogonal codes attaining the best minimum distance found so far.

cs.IT↗

arXiv2026

Shortest self-orthogonal and LCD embeddings of linear codes over Fq+uFq

This paper determines the exact lengths of shortest self-orthogonal and LCD embeddings of linear codes over $\mathbb{F}_q+u\mathbb{F}_q$. By decomposing Gram matrices over $\mathbb{F}_q+u\mathbb{F}_q$ into pairs of symmetric matrices over the finite field $\mathbb{F}_q$, the embedding problems are reduced to the congruence classification of symmetric and alternate matrices over finite fields. Complete formulas for the shortest self-orthogonal embedding length are obtained, with two distinct cases arising in both even and odd characteristic. We also show that every self-orthogonal code over $\mathbb{F}_q+u\mathbb{F}_q$ with nonzero free rank can be viewed as a shortest self-orthogonal embedding of another code. We use Witt theory to construct all shortest self-orthogonal embeddings. A complete characterization of shortest LCD embeddings is also established in terms of invertible and arbitrary matrices of prescribed sizes appended to a generator matrix. Examples of self-orthogonal and LCD embeddings with the largest minimum distance for the code considered are also presented, some of whose Gray images are optimal codes over $\mathbb{F}_q$.

cs.IT↗

arXiv2026

Cyclic codes over a commutative non-unitary ring of order 4

Let $I_2$ be the commutative non-unitary ring of order $4$ arising in the classification of Fine. In this paper, we investigate cyclic codes over $I_2$ through their associated residue and torsion codes over $\mathbb{F}_2$. We introduce the notions of twisted and untwisted cyclic codes and characterize cyclicity in terms of a compatibility condition involving the twist map and the cyclic shift. Connections between cyclic codes over $I_2$ and binary quasi-cyclic codes are established via Gray maps. In particular, we show that the Gray image of a cyclic code over $I_2$ is a binary quasi-cyclic code of index $2$. We also study duality properties of cyclic codes over $I_2$ and prove that the dual of a cyclic code is again cyclic. Finally, we classify permutation inequivalent cyclic codes over $I_2$ for lengths $n \le 7$ and determine various structural properties of these codes.

cs.IT↗

arXiv2026

Construction of self-orthogonal codes over a commutative non-unitary ring of order 25

Codes over non-unitary rings have been studied recently. In particular, codes over the commutative non-unitary ring $I_p$ (in the classification of Fine) of order $p^2$ where $p$ is a prime are being considered. For $p=2$ (resp. $p=3$), three categories of codes over $I_p$ have been studied: self-orthogonal codes, quasi self-dual codes, and self-dual codes over $I_p$. Using some related mass formulas and building-up constructions, classifications of these codes have been done up to the permutation equivalence (resp. the monomial equivalence) for certain small lengths. In this paper, we take the prime $p=5$ and consider the ring $I_5$. We introduce the notion of linear codes over $I_5$. We also define the same three categories of linear $I_5$-codes, study the structures of these $I_5$-codes and relate them to their associated residue and torsion codes. We classify the three categories of codes completely in lengths at most $4$ up to the monomial equivalence for a given type $\{ k_1 , k_2 \}$. Moreover, in the paper of Alahmadi et al. regarding the mass formula for self-orthogonal codes over $I_p$, mistakes in the classification of quasi self-dual codes over $I_5$ had been made such as incorrect automorphism group order of some codes or inconsistency with the mass formula for self-orthogonal codes over $I_p$ for length $n=2$ and type $\{ 1 , 0 \}$ and for length $n=3$ and type $\{ 1, 1 \}$. We correct and improve such results.

cs.IT↗

arXiv2026

Embedding linear codes over Z4 into self-orthogonal codes

The purpose of this paper is to investigate the self-orthogonal embedding problem for linear codes over Z4. We propose several tight bounds on the length of the shortest self-orthogonal embedding over Z4, and determine the exact shortest self-orthogonal embedding length under specific conditions. As an example satisfying these conditions, we establish the exact length of the shortest self-orthogonal embedding for the quaternary Preparata codes. Furthermore, to establish these results, we completely classify the exact length of the shortest doubly even self-orthogonal embedding for binary linear codes in every possible case. Finally, when the shortest self-orthogonal embedding length of a given free code over Z4 is equal to the shortest doubly even self-orthogonal embedding length of its residue code, we present an algorithm to construct all possible shortest self-orthogonal embeddings. With our algorithm, we found twelve linear codes over Z4 whose minimum Lee distances are higher than those of the Z4-linear codes in Aydins database.

cs.IT↗

arXiv2026

Symmetric Sudoku-Type Games from Perfect Codes

This paper presents a novel construction method for symmetric Sudoku-type games based on Lee distance perfect codes and diameter perfect codes. The proposed method utilizes the tiling property of these codes to define the structure of the subgrid constraints of Sudoku-type games. In this way, our games inherit the symmetric properties of Sudoku. We provide a detailed analysis of two small cases: a $5 \times 5$ Sudoku in $\mathbb{Z}_5^2$, and an $8 \times 8$ Sudoku in $\mathbb{Z}_8^2$. By defining equivalence relations via rigid motions, we provide a complete enumeration of valid grids, identifying 17 inequivalent solutions for $5\times 5$ Sudoku. For two different types of $8\times 8$ Sudoku, we characterize 232,735 and 304,014 inequivalent solutions, respectively. Furthermore, to verify practical playability, we implement a human-like solver that assesses the difficulty of the generated games. The analysis confirms that our $5\times5$ Sudoku games offer a balanced distribution of difficulty levels, ranging from Easy to Hard, making them a viable alternative to traditional $9 \times 9$ Sudoku.

math.CO↗

arXiv2026

Soohak: A Mathematician-Curated Benchmark for Evaluating Research-level Math Capabilities of LLMs

Following the recent achievement of gold-medal performance on the IMO by frontier LLMs, the community is searching for the next meaningful and challenging target for measuring LLM reasoning. Whereas olympiad-style problems measure step-by-step reasoning alone, research-level problems use such reasoning to advance the frontier of mathematical knowledge itself, emerging as a compelling alternative. Yet research-level math benchmarks remain scarce because such problems are difficult to source (e.g., Riemann Bench and FrontierMath-Tier 4 contain 25 and 50 problems, respectively). To support reliable evaluation of next-generation frontier models, we introduce Soohak, a 439-problem benchmark newly authored from scratch by 64 mathematicians. Soohak comprises two subsets. On the Challenge subset, frontier models including Gemini-3-Pro, GPT-5, and Claude-Opus-4.5 reach 30.4%, 26.4%, and 10.4% respectively, leaving substantial headroom, while leading open-weight models such as Qwen3-235B, GPT-OSS-120B, and Kimi-2.5 remain below 15%. Notably, beyond standard problem solving, Soohak introduces a refusal subset that probes a capability intrinsic to research mathematics: recognizing ill-posed problems and pausing rather than producing confident but unjustified answers. On this subset, no model exceeds 50%, identifying refusal as a new optimization target that current models do not directly address. To prevent contamination, the dataset will be publicly released in late 2026, with model evaluations available upon request in the interim.

cs.CL↗

arXiv2026

Formalizing building-up constructions of self-dual codes through isotropic lines in Lean

The purpose of this paper is two-fold. First, we show that, after a specified form isometry, the two-coordinate reduction in the binary Hilbert-symbol realization of Chinburg and Zhang is inverse to Kim's building-up construction, up to permutation equivalence. Second, for $q\equiv1\pmod4$, we develop a $q$-ary analogue of this reduction-and-extension mechanism. The identity $c^2=-1$ yields the isotropic line governing the split construction. For every fixed ordered pairing of the coordinates, we obtain a universal rank-$r$ boxed normal form, where $r$ is the dimension of the intersection with the product of these isotropic lines. Applications include optimal self-dual $[6,3,4]$ and $[8,4,4]$ codes over $\mathbb F_{5}$, optimal self-dual $[8,4,5]$ and $[10,5,6]$ codes over $\mathbb F_{13}$, and a self-dual $[12,6,6]$ code over $\mathbb F_{13}$. We also give an exact repeated boxed realization of self-dual $[18,9,8]$ and $[20,10,10]$ codes over $\mathbb F_{13}$, in which the split-boxed parent and its building-up child occur in one complete generator matrix. The algebraic core is formalized in Lean 4.

cs.IT↗

arXiv2026

Second order Recurrences, quadratic number fields and cyclic codes

Wall-Sun-Sun primes (shortly WSS primes) are defined as those primes $p$ such that the period of the Fibonacci recurrence is the same modulo $p$ and modulo $p^2.$ This concept has been generalized recently to certain second order recurrences whose characteristic polynomials admit as a zero the principal unit of $\mathbb{Q}(\sqrt{d}),$ for some integer $d>0.$ Primes of the latter type we call $WSS(d).$ They correspond to the case when $\mathbb{Q}(\sqrt{d})$ is not $p$-rational. For such a prime $p$ we study the weight distributions of the cyclic codes over $\mathbb{F}_p$ and $\mathbb{Z}_{p^2}$ whose check polynomial is the reciprocal of the said characteristic polynomial. Some of these codes are MDS (reducible case) or NMDS (irreducible case).

math.NT↗

arXiv2026

New bounds for codes over Gaussian integers based on the Mannheim distance

We study linear codes over Gaussian integers equipped with the Mannheim distance. We develop Mannheim-metric analogues of several classical bounds. We derive an explicit formula for the volume of Mannheim balls, which yields a sphere packing bound and constraints on the parameters of two-error-correcting perfect codes. We prove several other useful bounds, and exhibit families of codes meeting these bounds for some parameters, thereby showing that these bounds are tight. We also discuss self-dual codes over Gaussian integers and obtain upper bounds on their minimum Mannheim distance for certain parameter regions using a Mannheim version of the Macwilliams-type identity. Finally, we present decoding algorithms for codes over Gaussian integer residue rings. We give examples showing that certain errors which are not correctable under the Hamming metric become correctable under the Mannheim metric.

cs.IT↗

arXiv2026

Shortest LCD embeddings of binary, ternary and quaternary linear codes

In the recent years, there has been active research on self-orthogonal embeddings of linear codes since they yielded some optimal self-orthogonal codes. LCD codes have a trivial hull so they are counterparts of self-orthogonal codes. So it is a natural question whether one can embed linear codes into optimal LCD codes. To answer it, we first determine the number of columns to be added to a generator matrix of a linear code in order to embed the given code into an LCD code. Then we characterize all possible forms of shortest LCD embeddings of a linear code. As examples, we start from binary and ternary Hamming codes of small lengths and obtain optimal LCD codes with minimum distance 4. Furthermore, we find new ternary LCD codes with parameters including $[23, 4, 14]$, $[23, 5, 12]$, $[24, 6, 12]$, and $[25, 5, 14]$ and a new quaternary LCD $[21, 10, 8]$ code, each of which has minimum distance one greater than those of known codes. This shows that our shortest LCD embedding method is useful in finding optimal LCD codes over various fields.

cs.IT↗

arXiv2025

Shortest self-orthogonal embeddings of binary linear codes

There has been recent interest in the study of shortest self-orthogonal embeddings of binary linear codes, since many such codes are optimal self-orthogonal codes. Several authors have studied the length of a shortest self-orthogonal embedding of a given binary code $\mathcal C$, or equivalently, the minimum number of columns that must be added to a generator matrix of $\mathcal C$ to form a generator matrix of a self-orthogonal code. In this paper, we use properties of the hull of a linear code to determine the length of a shortest self-orthogonal embedding of any binary linear code. We focus on the examples of Hamming codes and Reed-Muller codes. We show that a shortest self-orthogonal embedding of a binary Hamming code is self-dual, and propose two algorithms to construct self-dual codes from Hamming codes $\mathcal H_r$. Using these algorithms, we construct a self-dual $[22, 11, 6]$ code, called the shortened Golay code, from the binary $[15, 11, 3]$ Hamming code $\mathcal H_4$, and construct a self-dual $[52, 26, 8]$ code from the binary $[31, 26, 3]$ Hamming code $\mathcal H_5$. We use shortest SO embeddings of linear codes to obtain many inequivalent optimal self-orthogonal codes of dimension $7$ and $8$ for several lengths. Four of the codes of dimension $8$ that we construct are codes with new parameters such as $[91, 8, 42],\, [98, 8, 46],\,[114, 8, 54]$, and $[191, 8, 94]$.

cs.IT↗

arXiv2025

How to Expand a Self-orthogonal Code

In this paper, we show how to expand Euclidean/Hermitian self-orthogonal code preserving their orthogonal property. Our results show that every $k$-dimension Hermitian self-orthogonal code is contained in a $(k+1)$-dimensional Hermitian self-orthogonal code. Also, for $k< n/2-1$, every $[n,k]$ Euclidean self-orthogonal code is contained in an $[n,k+1]$ Euclidean self-orthogonal code. Moreover, for $k=n/2-1$ and $p=2$, we can also fulfill the expanding process. But for $k=n/2-1$ and $p$ odd prime, the expanding process can be fulfilled if and only if an extra condition must be satisfied. We also propose two feasible algorithms on these expanding procedures.

cs.IT↗

arXiv2025

Galois equiangular tight frames from Galois self-dual codes

Greaves et al. (2022) extended frames over real or complex numbers to frames over finite fields. In this paper, we study the theory of frames over finite fields by incorporating the Galois inner products introduced by Fan and Zhang (2017), which generalize the Euclidean and Hermitian inner products. We define a class of frames, called Galois frames over finite fields, along with related notions such as Galois Gram matrices, Galois frame operators, and Galois equiangular tight frames (Galois ETFs). We also characterize when Galois self-dual codes induce Galois ETFs. Furthermore, we construct explicitly Galois ETFs from Galois self-dual constacyclic codes.

cs.IT↗

arXiv2024

Log-Concave Sequences in Coding Theory

We introduce the notion of logarithmically concave (or log-concave) sequences in Coding Theory. A sequence $a_0, a_1, \dots, a_n$ of real numbers is called log-concave if $a_i^2 \ge a_{i-1}a_{i+1}$ for all $1 \le i \le n-1$. A natural sequence of positive numbers in coding theory is the weight distribution of a linear code consisting of the nonzero values among $A_i$'s where $A_i$ denotes the number of codewords of weight $i$. We call a linear code log-concave if its nonzero weight distribution is log-concave. Our main contribution is to show that all binary general Hamming codes of length $2^r -1$ ($r=3$ or $r \ge 5$), the binary extended Hamming codes of length $2^r ~(r \ge 3)$, and the second order Reed-Muller codes $R(2, m)~ (m \ge 2)$ are all log-concave while the homogeneous and projective second order Reed-Muller codes are either log-concave, or 1-gap log-concave. Furthermore, we show that any MDS $[n, k]$ code over $\mathbb F_q$ satisfying $3 \leqslant k \leqslant n/2 +3$ is log-concave if $q \geqslant q_0(n, k)$ which is the larger root of a quadratic polynomial. Hence, we expect that the concept of log-concavity in coding theory will stimulate many interesting problems.

cs.IT↗

arXiv2024

Fuzzy linear codes based on nested linear codes

In this paper, we describe a correspondence between a fuzzy linear code and a family of nested linear codes. We also describe the arithmetic of fuzzy linear codes. As a special class of nested linear codes, we consider a family of nested self-orthogonal codes. A linear code is self-orthogonal if it is contained in its dual and self-dual if it is equal to its dual. We introduce a definition of fuzzy self-dual or self-orthogonal codes which include classical self-dual or self-orthogonal codes. As examples, we construct several interesting classes of fuzzy linear codes including fuzzy Hamming codes, fuzzy Golay codes, and fuzzy Reed-Muller codes. We also give a general decoding algorithm for fuzzy linear codes.

cs.IT↗

arXiv2024

Expanding self-orthogonal codes over a ring $\Z_4$ to self-dual codes and unimodular lattices

Self-dual codes have been studied actively because they are connected with mathematical structures including block designs and lattices and have practical applications in quantum error-correcting codes and secret sharing schemes. Nevertheless, there has been less attention to construct self-dual codes from self-orthogonal codes with smaller dimensions. Hence, the main purpose of this paper is to propose a way to expand any self-orthogonal code over a ring $\Z_4$ to many self-dual codes over $\Z_4$. We show that all self-dual codes over $\Z_4$ of lengths $4$ to $8$ can be constructed this way. Furthermore, we have found five new self-dual codes over $\Z_4$ of lengths $27, 28, 29, 33,$ and $34$ with the highest Euclidean weight $12$. Moreover, using Construction $A$ applied to our new Euclidean-optimal self-dual codes over $\Z_4$, we have constructed a new odd extremal unimodular lattice in dimension 34 whose kissing number was not previously known.

cs.CR↗