Jon-Lark Kim

• Sogang University

There are four commutative unital rings of order four. Among them, we consider the ring \(R=\mathbb {F}_{2}+u\mathbb {F}_{2}= \left\{ 0,1,u,\bar{u}=u+1\right\} \) where \(u^2=0\), which is a commutative ring with characteristic 2. In this paper, we study linear complementary dual (LCD) codes over the ring R. We first define \({\text{ LCD }}[n,k]_{R...

Projective Reed–Muller codes are constructed from the family of projective hypersurfaces of a fixed degree over a finite field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{docu...

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...

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 dimensi...

Triorthogonal matrices were introduced in Quantum Information Theory in connection with distillation of magic states (Bravyi and Haah (2012)). We give an algorithm to construct binary triorthogonal matrices from binary self-dual codes. Further, we generalize to this setting the classical coding techniques of shortening and extending. We also give s...

Triorthogonal matrices were introduced in quantum information theory in connection with distillation of magic states (Bravyi and Haah in Phys Rev A 86:052329, 2012). We give an algorithm to construct binary triorthogonal matrices from binary self-dual codes. Further, we generalize to this setting the classical coding techniques of shortening and ex...

Projective Reed-Muller codes are constructed from the family of projective hypersurfaces of a fixed degree over a finite field $\F_q$. We consider the relationship between projective Reed-Muller codes and their duals. We determine when these codes are self-dual, when they are self-orthogonal, and when they are LCD. We then show that when $q$ is suf...

In this paper, we introduce a Genetic Algorithm based Upper Confidence Bound (GA-UCB), an innovative hybrid genetic algorithm integrating Multi-Armed Bandit (MAB). It effectively addresses the challenges of solving large and intricate Sudoku puzzles, thus overcoming the constraints of traditional genetic algorithms. In GA-UCB, reinforcement learnin...

Modular strongly regular graphs have been introduced by Greaves et al. (Linear Algebra Appl 639:50–80, 2022). We show that a related class of isodual codes is asymptotically good. Equiangular tight frames over finite fields also introduced by the same authors in 2022 are shown here to connect with self-dual codes. We give several examples of equian...

Array-designed codes are capable of correcting row and column deletions. In this paper, we introduce array-designed reversible and complementary codes over the finite field GF(4), which can correct row deletions and column deletions errors. Hence our codes can be useful for DNA codes. We also propose a construction method of array-designed reversib...

The authors study the binary codes spanned by the adjacency matrices of the strongly regular graphs (SRGs) on at most two hundred vertices whose existence is unknown. The authors show that in length less than one hundred they cannot be cyclic, except for the exceptions of the SRGs of parameters (85, 42, 20, 21) and (96, 60, 38, 36). In particular,...

The purpose of this paper is to solve the two conjectures on the largest minimum distance d_so ( n , 5) of a binary self-orthogonal [ n , 5] code proposed by Kim and Choi (2022). The determination of d_so ( n, k ) has been a fundamental and difficult problem in coding theory because there are too many binary self-orthogon...

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 eq...

The main purpose of this paper is to classify Type I (self-dual) codes over \({\mathbb {F}}_{4}+u{\mathbb {F}}_{4}\) where \(u^2=0\). By the standard form of generator matrix or a building-up construction, we classify all Type I codes of lengths up to 6. We confirm that our classification is correct by checking with the mass formula for Type I code...

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 eq...

The purpose of this paper is two-fold. First, we characterize the existence of binary self-orthogonal codes meeting the Griesmer bound by employing Solomon-Stiffler codes and some related residual codes. Second, using such a characterization, we determine the exact value of $d_{so}(n,7)$ except for five special cases and the exact value of $d_{so}(...

The hull of a linear code C is the intersection of C with its dual. To the best of our knowledge, there are very few constructions of binary linear codes with the hull dimension \(\ge 2\) except for self-orthogonal codes. We propose a building-up construction to obtain a plenty of binary \([n+2, k+1]\) codes with hull dimension \(\ell , \ell +1\),...

The hull of a linear code over finite fields is the intersection of the code and its dual. Linear codes with small hulls have been widely studied due to their applications in computational complexity and information protection. In this paper, we study some properties of binary linear codes with one-dimensional hull, and establish their relation wit...

A pair of linear codes (C, D) of length n over Fq is called a linear complementary pair (LCP) if their direct sum yields the full space Fqn. By a result of Carlet et al. (2019), the best security parameters of binary LCPs of codes are left open. Motivated by this, we study binary LCPs of codes. We describe a sufficient condition for binary LCPs of...

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 matri...

The hull of a linear code $C$ is the intersection of $C$ with its dual. To the best of our knowledge, there are very few constructions of binary linear codes with the hull dimension $\ge 2$ except for self-orthogonal codes. We propose a building-up construction to obtain a plenty of binary $[n+2, k+1]$ codes with hull dimension $\ell, \ell +1$, or...

The purpose of this paper is to solve the two conjectures on the largest minimum distance $d_{so}(n,5)$ of a binary self-orthogonal $[n,5]$ code proposed by Kim and Choi (IEEE Trans. Inf. Theory, 2022). The determination of $d_{so}(n,k)$ has been a fundamental and difficult problem in coding theory because there are too many binary self-orthogonal...

Kim and Ohk (2022) showed that binary self-orthogonal [n, k] codes for various dimensions k can be useful for the construction of DNA codes based on quasi self-dual codes over a noncommutative nonunital ring E with four elements. However, there is few classification of binary self-orthogonal codes with dimension \(\ge 6\). In this paper, we complet...

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 in Pinto et al (eds) Coding theory and applications, Springer, CIMSMS, Berlin, 2015). Additive codes over \(\mathbb {F}_4\) are \(\ma...

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...

Steganography is the science of communicating a secret message by hiding it in a cover object. The well known F5 algorithm is based on binary Hamming codes in Hamming graph. It has been an interesting research problem to construct other steganographic schemes from mathematically structured graphs. In this paper, we construct steganographic schemes...

Let I be the commutative non-unital ring of order 4 defined by generators and relations. I=a,b∣2a=2b=0,a2=b,ab=0.Alahmadi et al. have classified QSD codes, Type IV codes (QSD codes with even weights) and quasi-Type IV codes (QSD codes with even torsion code) over I up to lengths n=6, and suggested two building-up methods for constructing QSD codes....

We study LCD (linear complementary dual) and ACD (additive complementary dual) codes over a noncommutative non-unital ring E with four elements. This is the first attempt to construct LCD codes over a noncommutative non-unital ring. We show that free LCD codes over E are directly related to binary LCD codes. We introduce ACD codes over E. They incl...

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 extends this result for $k=5$ and $6$ by proposing a new method related to a special matrix, called the self-orthogonalit...

This paper gives new methods of constructing 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 ever...

There is a local ring E of order 4, without identity for the multiplication, defined by generators and relations as \(E=\langle a,b \mid 2a=2b=0,\, a^2=a,\, b^2=b,\,ab=a,\, ba=b\rangle .\) We study a special construction of self-orthogonal codes over E, based on combinatorial matrices related to two-class association schemes, Strongly Regular Graph...

In this paper, we describe a new type of DNA codes over two noncommutative rings E and F of order four with characteristic 2. Our DNA codes are based on quasi self-dual codes over E and F. Using quasi self-duality, we can describe fixed GC-content constraint weight distributions and reverse-complement constraint minimum distributions of those codes...

We introduce a consistent and efficient method to construct self-dual codes over GF(q) using symmetric matrices and eigenvectors from a self-dual code over GF(q) of smaller length where q ≡ 1 (mod 4). Using this method, which is called a ‘symmetric building-up’ construction, we improve the bounds of the best-known minimum weights of self-dual codes...

There is a local ring $E$ of order $4,$ without identity for the multiplication, defined by generators and relations as $E=\langle a,b \mid 2a=2b=0,\, a^2=a,\, b^2=b,\,ab=a,\, ba=b\rangle.$ We study a special construction of self-orthogonal codes over $E,$ based on combinatorial matrices related to two-class association schemes, Strongly Regular Gr...

We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an algorithmic method to embed a given binary k-dimensional linear code C ( k = 3,4) into a self-orthogonal code of the...

Dual-Ouroboros (Gaborit et al. in Adv Math Commun, 2019. 10.3934/amc.2020021) is a code-based public-key encryption scheme which is a modification of McNie and a dual version of Ouroboros-R. In this paper, we present a modification of Dual-Ouroboros, using a class of rank metric codes called Gabidulin codes. By using Gabidulin codes, we eliminate t...

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...

DNA codes based on error-correcting codes have been successful in DNA-based computation and storage. Since there are four nucleobases in DNA, two well known algebraic structures such as the finite field $GF(4)$ and the integer modular ring $\mathbb{Z}_4$ have been used. However, due to various possibilities of DNA sequences, it is natural to ask wh...

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...

Matrix codes over a finite field \({\mathbb {F}}_q\) are linear codes defined as subspaces of the vector space of \(m \times n\) matrices over \({\mathbb {F}}_q\). In this paper, we show how to obtain self-dual matrix codes from a self-dual matrix code of smaller size using a method we call the building-up construction. We show that every self-dual...

We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an algorithmic method to embed a given binary $k$-dimensional linear code $\mathcal{C}$ ($k = 2,3,4$) into a self-orthogo...

Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes, such as cyclic codes, Reed–Solomon codes, and Reed–Muller codes, have nice decoding algorithms. However, many optimal linear codes do not have an efficient decoding algorithm except for the general syndrome d...

Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes such as cyclic codes, Reed-Solomon codes, and Reed-Muller codes have nice decoding algorithms. However, many optimal linear codes do not have an efficient decoding algorithm except for the general syndrome dec...

The main purpose of this paper is to give a new and general way to obtain steganographic schemes from perfect codes on Cayley graphs, motivated by the F5 algorithm based on binary Hamming codes. We obtain the steganography based on perfect Hamming codes as a special case and also give various equivalent conditions for the existence of a perfect cod...

In this paper, we suggest a code-based public key encryption scheme, called McNie. McNie is a hybrid version of the McEliece and Niederreiter cryptosystems and its security is reduced to the hard problem of syndrome decoding. The public key involves a random generator matrix which is also used to mask the code used in the secret key. This makes the...

McNie is a code-based public key encryption scheme submitted as a candidate to the NIST Post-Quantum Cryptography standardization. In this paper, we present McNie2-Gabidulin, an improvement of McNie. By using Gabidulin code, we eliminate the decoding failure, which is one of the limitations of the McNie public key cryptosystem that uses LRPC codes....

Choi Seok-Jeong, a Korean mathematician, studied Latin squares at least 60 years earlier than Euler. He introduced a pair of orthogonal Latin squares of order 9 in his book called Koo-Soo-Ryak (also pronounced as Gusuryak). Interestingly, his two orthogonal non-diagonal Latin squares produce a magic square of order 9, whose theoretical reason was n...

This presentation is related to the paper: S. T. Dougherty, J.-L. Kim, B. Ö., L. Sok and P. Solé, “The combinatorics of LCD codes: Linear programming bound and orthogonal matrices”, Int. J. Inf. and Coding Theory, vol. 4, no. 2/3, 116-128, 2017. The slides clarify some short proofs in Sections 3 and 4. An explicit construction is given for the pr...

We study the construction of quasi-cyclic self-dual codes, especially of binary cubic ones. We consider the binary quasi-cyclic codes of length \(3\ell \) with the algebraic approach of Ling and Solé (IEEE Trans Inf Theory 47(7):2751–2760, 2001. doi:10.1109/18.959257). In particular, we improve the previous results by constructing 1 new binary [54,...

A linear code with a complementary dual (or LCD code) is defined to be a linear code $C$ whose dual code $C^{\perp}$ satisfies $C \cap C^{\perp}$= $\left\{ \mathbf{0}\right\} $. Let $LCD{[}n,k{]}$ denote the maximum of possible values of $d$ among $[n,k,d]$ binary LCD codes. We give exact values of $LCD{[}n,k{]}$ for $1 \le k \le n \le 12$. We also...

In this paper, we propose a variant of the McEliece cryptosystem, called LRPC-Kronecker cryptosystem. LRPC-Kronecker product codes are LRPC codes with higher rank and better error-correction capability. For this, we introduce a new decoding algorithm using blocks which has lower decoding complexity and higher error-correction capability compared to...

We examine various dualities over the fields of even orders, giving new dualities forFadditive codes. We relate the MacWilliams relations and the duals of F22s codes for these various dualities. We study self-dual codes with respect to these dualities and prove that any subgroup of order 2s of the additive group is a self-dual code with respect to...

We study the construction of quasi-cyclic self-dual codes, especially of binary cubic ones. We consider the binary quasi-cyclic codes of length 3\ell with the algebraic approach of [9]. In particular, we improve the previous results by constructing 1 new binary [54, 27, 10], 6 new [60, 30, 12] and 50 new [66, 33, 12] cubic self-dual codes. We conje...

As far as we know, there is no decoding algorithm of any binary self-dual [40, 20, 8] code except for the syndrome decoding applied to the code directly. This syndrome decoding for a binary self-dual [40, 20, 8] code is not efficient in the sense that it cannot be done by hand due to a large syndrome table. The purpose of this paper is to give two...

A secret sharing scheme (SSS) was introduced by Shamir in 1979 using polynomial interpolation. Later it turned out that it is equivalent to an SSS based on a Reed-Solomon code. SSSs based on linear codes have been studied by many researchers. However there is little research on SSSs based on additive codes. In this paper, we study SSSs based on add...

As far as we know, there is no decoding algorithm of any binary self-dual $[40, 20, 8]$ code except for the syndrome decoding applied to the code directly. This syndrome decoding for a binary self-dual $[40,20,8]$ code is not efficient in the sense that it cannot be done by hand due to a large syndrome table. The purpose of this paper is to give tw...

A linear code with a complementary dual (or LCD code) is defined to be a linear code $C$ whose dual code $C^{\perp}$ satisfies $C \cap C^{\perp}$= $\left\{ \mathbf{0}\right\} $. Let $LCD{[}n,k{]}$ denote the maximum of possible values of $d$ among $[n,k,d]$ binary LCD codes. We give exact values of $LCD{[}n,k{]}$ for $1 \le k \le n \le 12$. We also...

A secret sharing scheme (SSS) was introduced by Shamir in 1979 using polynomial interpolation. Later it turned out that it is equivalent to an SSS based on a Reed–Solomon code. SSSs based on linear codes have been studied by many researchers. However there is little research on SSSs based on additive codes. In this paper, we study SSSs based on add...

The rank over a finite field of the adjacency matrix of a directed strongly regular graph is studied, with some applications to the construction of linear codes. Three techniques are used: code orthogonality, adjacency matrix determinant, and adjacency matrix spectrum.

Linear complementary dual (LCD) codes 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 Rk . We give a linear programming bound on the largest size of an LCD code of given length and minimum distance. We ma...

Crnković (2014) introduced a self-orthogonal [ ] code and a self-dual [ ] code over the finite field arising from orbit matrices for Menon designs, for every prime power , where and a prime dividing . He showed that if is a prime and , where is a non-negative integer, then the self-dual [ ] code over is equivalent to a Pless symmetry code. However...

This book, ‘Linear Algebra with Sage’, has two goals. The first goal is to explain Linear Algebra with the help of Sage. Sage is one of the most popular computer algebra system(CAS). Sage is a free and user-friendly software. Whenever the Sage codes are possible, we illustrate examples with Sage codes. The second goal is to make the book accessibl...