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Jon-Lark Kim

Jon-Lark Kim appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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Information Theorymath.ITmath.CODiscrete MathematicsComputation and LanguageCryptography and Securitycs.CYmath.ACmath.NT

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Published work

15 published item(s)

preprint2026arXiv

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.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Additive complementary dual codes over $\F_4$

A linear code is linear complementary dual (LCD) if it meets its dual trivially. LCD codes have been a hot topic recently due to Boolean masking application in the security of embarked electronics (Carlet and Guilley, 2014). Additive codes over $\F_4$ are $\F_4$-codes that are stable by codeword addition but not necessarily by scalar multiplication. An additive code over $\F_4$ is additive complementary dual (ACD) if it meets its dual trivially. The aim of this research is to study such codes which meet their dual trivially. All the techniques and problems used to study LCD codes are potentially relevant to ACD codes. Interesting constructions of ACD codes from binary codes are given with respect to the trace Hermitian and trace Euclidean inner product. The former product is relevant to quantum codes.

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preprint2022arXiv

Extending binary linear codes to self-orthogonal codes

Kim et al. (2021) gave a method to embed a given binary $[n,k]$ code $\mathcal{C}$ $(k = 3, 4)$ into a self-orthogonal code of the shortest length which has the same dimension $k$ and minimum distance $d' \ge d(\mathcal{C})$. We extend this result by proposing a new method related to a special matrix, called the self-orthogonality matrix $SO_k$, obtained by shortening a Reed-Muller code $\mathcal R(2,k)$. Using this approach, we can extend binary linear codes to many optimal self-orthogonal codes of dimensions $5$ and $6$. Furthermore, we partially disprove the conjecture (Kim et al. (2021)) by showing that if $31 \le n \le 256$ and $n\equiv 14,22,29 \pmod{31}$, then there exist optimal $[n,5]$ codes which are self-orthogonal. We also construct optimal self-orthogonal $[n,6]$ codes when $41 \le n \le 256$ satisfies $n \ne 46, 54, 61$ and $n \not\equiv 7, 14, 22, 29, 38, 45, 53, 60 \pmod{63}$.

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preprint2021arXiv

An improved upper bound on self-dual codes over finite fields $GF(11), GF(19)$, and $GF(23)$

This paper gives new methods of constructing {\it symmetric self-dual codes} over a finite field $GF(q)$ where $q$ is a power of an odd prime. These methods are motivated by the well-known Pless symmetry codes and quadratic double circulant codes. Using these methods, we construct an amount of symmetric self-dual codes over $GF(11)$, $GF(19)$, and $GF(23)$ of every length less than 42. We also find 153 {\it new} self-dual codes up to equivalence: they are $[32, 16, 12]$, $[36, 18, 13]$, and $[40, 20,14]$ codes over $GF(11)$, $[36, 18, 14]$ and $[40, 20, 15]$ codes over $GF(19)$, and $[32, 16, 12]$, $[36, 18, 14]$, and $[40, 20, 15]$ codes over $GF(23)$. They all have new parameters with respect to self-dual codes. Consequently, we improve bounds on the highest minimum distance of self-dual codes, which have not been significantly updated for almost two decades.

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preprint2021arXiv

Self-dual codes over $GF(q)$ with symmetric generator matrices

We introduce a consistent and efficient method to construct self-dual codes over $GF(q)$ with symmetric generator matrices from a self-dual code over $GF(q)$ of smaller length where $q \equiv 1 \pmod 4$. Using this method, we improve the best-known minimum weights of self-dual codes, which have not significantly improved for almost two decades. We focus on a class of self-dual codes, including double circulant codes. Using our method, called a `symmetric building-up' construction, we obtain many new self-dual codes over $GF(13)$ and $GF(17)$ and improve the bounds of best-known minimum weights of self-dual codes of lengths up to 40. Besides, we compute the minimum weights of quadratic residue codes that were not known before. These are: a [20,10,10] QR self-dual code over $GF(23)$, two [24,12,12] QR self-dual codes over $GF(29)$ and $GF(41)$, and a [32,12,14] QR self-dual codes over $GF(19)$. They have the highest minimum weights so far.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices

Linear Complementary Dual codes (LCD) are binary linear codes that meet their dual trivially. We construct LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes over the family of rings $R_k$. We give a linear programming bound on the largest size of an LCD code of given length and minimum distance. We make a table of lower bounds for this combinatorial function for modest values of the parameters.

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preprint2014arXiv

Higher-order CIS codes

We introduce {\bf complementary information set codes} of higher-order. A binary linear code of length $tk$ and dimension $k$ is called a complementary information set code of order $t$ ($t$-CIS code for short) if it has $t$ pairwise disjoint information sets. The duals of such codes permit to reduce the cost of masking cryptographic algorithms against side-channel attacks. As in the case of codes for error correction, given the length and the dimension of a $t$-CIS code, we look for the highest possible minimum distance. In this paper, this new class of codes is investigated. The existence of good long CIS codes of order $3$ is derived by a counting argument. General constructions based on cyclic and quasi-cyclic codes and on the building up construction are given. A formula similar to a mass formula is given. A classification of 3-CIS codes of length $\le 12$ is given. Nonlinear codes better than linear codes are derived by taking binary images of $\Z_4$-codes. A general algorithm based on Edmonds' basis packing algorithm from matroid theory is developed with the following property: given a binary linear code of rate $1/t$ it either provides $t$ disjoint information sets or proves that the code is not $t$-CIS. Using this algorithm, all optimal or best known $[tk, k]$ codes where $t=3, 4, \dots, 256$ and $1 \le k \le \lfloor 256/t \rfloor$ are shown to be $t$-CIS for all such $k$ and $t$, except for $t=3$ with $k=44$ and $t=4$ with $k=37$.

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preprint2014arXiv

Multiply Constant-Weight Codes and the Reliability of Loop Physically Unclonable Functions

We introduce the class of multiply constant-weight codes to improve the reliability of certain physically unclonable function (PUF) response. We extend classical coding methods to construct multiply constant-weight codes from known $q$-ary and constant-weight codes. Analogues of Johnson bounds are derived and are shown to be asymptotically tight to a constant factor under certain conditions. We also examine the rates of the multiply constant-weight codes and interestingly, demonstrate that these rates are the same as those of constant-weight codes of suitable parameters. Asymptotic analysis of our code constructions is provided.

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preprint2012arXiv

A new class of codes for Boolean masking of cryptographic computations

We introduce a new class of rate one-half binary codes: {\bf complementary information set codes.} A binary linear code of length $2n$ and dimension $n$ is called a complementary information set code (CIS code for short) if it has two disjoint information sets. This class of codes contains self-dual codes as a subclass. It is connected to graph correlation immune Boolean functions of use in the security of hardware implementations of cryptographic primitives. Such codes permit to improve the cost of masking cryptographic algorithms against side channel attacks. In this paper we investigate this new class of codes: we give optimal or best known CIS codes of length $<132.$ We derive general constructions based on cyclic codes and on double circulant codes. We derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all be classified in small lengths $\le 12$ by the building up construction. Some nonlinear permutations are constructed by using $\Z_4$-codes, based on the notion of dual distance of an unrestricted code.

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preprint2012arXiv

An Efficient Construction of Self-Dual Codes

We complete the building-up construction for self-dual codes by resolving the open cases over $GF(q)$ with $q \equiv 3 \pmod 4$, and over $\Z_{p^m}$ and Galois rings $\GR(p^m,r)$ with an odd prime $p$ satisfying $p \equiv 3 \pmod 4$ with $r$ odd. We also extend the building-up construction for self-dual codes to finite chain rings. Our building-up construction produces many new interesting self-dual codes. In particular, we construct 945 new extremal self-dual ternary $[32,16,9]$ codes, each of which has a trivial automorphism group. We also obtain many new self-dual codes over $\mathbb Z_9$ of lengths $12, 16, 20$ all with minimum Hamming weight 6, which is the best possible minimum Hamming weight that free self-dual codes over $\Z_9$ of these lengths can attain. From the constructed codes over $\mathbb Z_9$, we reconstruct optimal Type I lattices of dimensions $12, 16, 20,$ and 24 using Construction $A$; this shows that our building-up construction can make a good contribution for finding optimal Type I lattices as well as self-dual codes. We also find new optimal self-dual $[16,8,7]$ codes over GF(7) and new self-dual codes over GF(7) with the best known parameters $[24,12,9]$.

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preprint2012arXiv

Classification of Binary Self-Dual [48,24,10] Codes with an Automorphism of Odd Prime Order

The purpose of this paper is to complete the classification of binary self-dual [48,24,10] codes with an automorphism of odd prime order. We prove that if there is a self-dual [48, 24, 10] code with an automorphism of type p-(c,f) with p being an odd prime, then p=3, c=16, f=0. By considering only an automorphism of type 3-(16,0), we prove that there are exactly 264 inequivalent self-dual [48, 24, 10] codes with an automorphism of odd prime order, equivalently, there are exactly 264 inequivalent cubic self-dual [48, 24, 10] codes.

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preprint2012arXiv

Computational Results of Duadic Double Circulant Codes

Quadratic residue codes have been one of the most important classes of algebraic codes. They have been generalized into duadic codes and quadratic double circulant codes. In this paper we introduce a new subclass of double circulant codes, called {\em{duadic double circulant codes}}, which is a generalization of quadratic double circulant codes for prime lengths. This class generates optimal self-dual codes, optimal linear codes, and linear codes with the best known parameters in a systematic way. We describe a method to construct duadic double circulant codes using 4-cyclotomic cosets and give certain duadic double circulant codes over $\mathbb F_2, \mathbb F_3, \mathbb F_4, \mathbb F_5$, and $\mathbb F_7$. In particular, we find a new ternary self-dual $[76,38,18]$ code and easily rediscover optimal binary self-dual codes with parameters $[66,33,12]$, $[68,34,12]$, $[86,43,16]$, and $[88,44,16]$ as well as a formally self-dual binary $[82,41,14]$ code.

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preprint2012arXiv

Construction of quasi-cyclic self-dual codes

There is a one-to-one correspondence between $\ell$-quasi-cyclic codes over a finite field $\mathbb F_q$ and linear codes over a ring $R = \mathbb F_q[Y]/(Y^m-1)$. Using this correspondence, we prove that every $\ell$-quasi-cyclic self-dual code of length $m\ell$ over a finite field $\mathbb F_q$ can be obtained by the {\it building-up} construction, provided that char $(\mathbb F_q)=2$ or $q \equiv 1 \pmod 4$, $m$ is a prime $p$, and $q$ is a primitive element of $\mathbb F_p$. We determine possible weight enumerators of a binary $\ell$-quasi-cyclic self-dual code of length $p\ell$ (with $p$ a prime) in terms of divisibility by $p$. We improve the result of [3] by constructing new binary cubic (i.e., $\ell$-quasi-cyclic codes of length $3\ell$) optimal self-dual codes of lengths $30, 36, 42, 48$ (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When $m=5$, we obtain a new 8-quasi-cyclic self-dual $[40, 20, 12]$ code over $\mathbb F_3$ and a new 6-quasi-cyclic self-dual $[30, 15, 10]$ code over $\mathbb F_4$. When $m=7$, we find a new 4-quasi-cyclic self-dual $[28, 14, 9]$ code over $\mathbb F_4$ and a new 6-quasi-cyclic self-dual $[42,21,12]$ code over $\mathbb F_4$.

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preprint2012arXiv

Optimum Subcodes of Self-Dual Codes and Their Optimum Distance Profiles

Binary optimal codes often contain optimal or near-optimal subcodes. In this paper we show that this is true for the family of self-dual codes. One approach is to compute the optimum distance profiles (ODPs) of linear codes, which was introduced by Luo, et. al. (2010). One of our main results is the development of general algorithms, called the Chain Algorithms, for finding ODPs of linear codes. Then we determine the ODPs for the Type II codes of lengths up to 24 and the extremal Type II codes of length 32, give a partial result of the ODP of the extended quadratic residue code $q_{48}$ of length 48. We also show that there does not exist a $[48,k,16]$ subcode of $q_{48}$ for $k \ge 17$, and we find a first example of a doubly-even self-complementary $[48, 16, 16]$ code.

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preprint2011arXiv

Classification of extremal and $s$-extremal binary self-dual codes of length 38

In this paper we classify all extremal and $s$-extremal binary self-dual codes of length 38. There are exactly 2744 extremal $[38,19,8]$ self-dual codes, two $s$-extremal $[38,19,6]$ codes, and 1730 $s$-extremal $[38,19,8]$ codes. We obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self-dual codes of length 36, and from a generalization of this recursive algorithm for the shadow. The classification of $s$-extremal $[38,19,6]$ codes permits to achieve the classification of all $s$-extremal codes with d=6.

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