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POSETS ADMITTING THE LINEARITY OF ISOMETRIES
현종윤,김정진,김상목 대한수학회 2015 대한수학회보 Vol.52 No.3
In this paper, we deal with a characterization of the posets with the property that every poset isometry of Fn q fixing the origin is a linear map. We say such a poset to be admitting the linearity of isometries. We show that a poset P admits the linearity of isometries over Fn q if and only if P is a disjoint sum of chains of cardinality 2 or 1 when q = 2, or P is an anti-chain otherwise.
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Characterization of certain types of $r$-plateaued functions
현종윤,이정연,이윤진 대한수학회 2018 대한수학회지 Vol.55 No.6
We study a subclass of $p$-ary functions in $n$ variables, denoted by ${\mathcal A}_n$, which is a collection of $p$-ary functions in $n$ variables satisfying a certain condition on the exponents of its monomial terms. Firstly, we completely classify all $p$-ary $(n-1)$-plateaued functions in $n$ variables by proving that every $(n-1)$-plateaued function should be contained in $\mathcal{A}_n$. Secondly, we prove that if $f$ is a $p$-ary $r$-plateaued function contained in ${\mathcal A}_n$ with $\deg{f} > 1+\frac{n-r}{4}(p-1)$, then the highest degree term of $f$ is only a single term. Furthermore, we prove that there is no $p$-ary $r$-plateaued function in ${\mathcal A}_n$ with maximum degree $(p-1)\frac{n-r}{2}+1$. As application, we partially classify all $(n-2)$-plateaued functions in ${\mathcal A}_n$ when $p=3,5,$ and $7$, and $p$-ary bent functions in ${\mathcal A}_2$ are completely classified for the cases $p=3$ and $5$.
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Construction of two- or three-weight binary linear codes from Vasil'ev codes
현종윤,Jaeseon Kim 대한수학회 2021 대한수학회지 Vol.58 No.1
The set $D$ of column vectors of a generator matrix of a linear code is called a defining set of the linear code. In this paper we consider the problem of constructing few-weight (mainly two- or three-weight) linear codes from defining sets. It can be easily seen that we obtain an one-weight code when we take a defining set to be the nonzero codewords of a linear code. Therefore we have to choose a defining set from a non-linear code to obtain two- or three-weight codes, and we face the problem that the constructed code contains many weights. To overcome this difficulty, we employ the linear codes of the following form: Let $D$ be a subset of $\mathbb{F}_2^n$, and $W$ (resp.~$V$) be a subspace of $\mathbb{F}_2$ (resp.~$\mathbb{F}_2^n$). We define the linear code $\mathcal{C}_D(W; V)$ with defining set $D$ and restricted to $W, V$ by \[ \mathcal{C}_D(W; V) = \{(s+u\cdot x)_{x\in D^*} \,|\, s\in W, u\in V\}. \] We obtain two- or three-weight codes by taking $D$ to be a Vasil'ev code of length $n=2^m-1 (m \geq 3)$ and a suitable choices of $W$. We do the same job for $D$ being the complement of a Vasil'ev code. The constructed few-weight codes share some nice properties. Some of them are optimal in the sense that they attain either the Griesmer bound or the Grey-Rankin bound. Most of them are minimal codes which, in turn, have an application in secret sharing schemes. Finally we obtain an infinite family of minimal codes for which the sufficient condition of Ashikhmin and Barg does not hold.
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Inversion of the classical Radon transform on $\mathbb{Z}^n_p$
조영덕,현종윤,문성환 대한수학회 2018 대한수학회보 Vol.55 No.6
The Radon transform introduced by J. Radon in 1917 is the integral transform which is widely applicable to tomography. Here we study the discrete version of the Radon transform. More precisely, when $\mathcal{C}(\mathbb{Z}^n_p)$ is the set of complex-valued functions on $\mathbb{Z}^n_p$. We completely determine the subset of $\mathcal{C}(\mathbb{Z}^n_p)$ whose elements can be recovered from its Radon transform on $\mathbb{Z}^n_p$.
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A NOTE ON AN ASYMPTOTIC FUNCTION
Ahn, Sung Hum,Hyun, Jong Yun 東國大學校 1998 東國論叢 Vol.37 No.-
함수 δ_I(J, K^s, m)을 정의하고 이 함수를 이용하여 만약 ideal J가 ideal K의 reduction인 경우 임의의 ideal I에 대하여 0≤α_I(K, m)-α_I(J, m)α≤r_J(K) 임을 보였다. 또한 임의의 ideal I에 대하여 ◁그림 삽입▷(원문을 참조하세요)이 존재할 필요충분조건은 ◁그림 삽입▷(원문을 참조하세요) 이 존재한다는 것임을 보이고, 그 극한값 ◁그림 삽입▷(원문을 참조하세요) 이 존재하는 경우 ◁그림 삽입▷(원문을 참조하세요) 임을 보였다. α_I(J, m)의 응용으로서, 만약 ideal J가 ideal K의 reduction의 경우, J가 regular일 필요충분조건이 K가 regular라는 것을 보였다. In this paper, we introduce the function, δ_1(J, K^2, m) and ahow that if J is a reduction of K then, 0≤α_1(K, m)-α_I(J, m)≤r_J(K) for any ideal I of R. We also show that if J is a reduction of an ideal K then, for any ideal I of R, ◁그림 삽입▷(원문을 참조하세요) exists if and only if ◁그림 삽입▷(원문을 참조하세요) exists. In particular, if the limit exists then ◁그림 삽입▷(원문을 참조하세요) As an application of α_I(J, m). we give another proof to show that if J is a reduction of K then, J is regular if and only if K is regular.
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NEW METRICS ARISING FROM MATROIDS OF LINEAR CODES
김동찬,김대산,현종윤 장전수학회 2013 Proceedings of the Jangjeon mathematical society Vol.16 No.2
In this paper, from each linear code in Fnq, we construct two metrics, the one called “the metric associated to it” and the other “the symmetrized metric associated to it.” They are respectively defined interms of the rank function and the rank and corank functions of the matroid of the linear code. Such metrics constructed from linear codes give rise to abundant new metrics on the space Fnq. We investigate their properties and illustrate our results with some examples.
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김대산,D. C. Kim,현종윤 장전수학회 2012 Advanced Studies in Contemporary Mathematics Vol.22 No.1
In 1991, Wei introduced generalized minimum Hamming weights for linear codes and showed their monotonicity and duality. Re-cently, several authors extended these results to the case of generalized minimum poset weights by using different methods. Here, we would like to prove the duality by using matroid theory. This gives yet another and very simple proof of it. In particular, our argument will make it clear that the duality follows from the well-known relation between the rank function and the corank function of a matroid. In addition, we derive the weight distributions of linear MDS and Near-MDS poset codes in the same spirit.
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