• 충청수학회지
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• 제16권1호
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• pp.61-71
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• 2003

We obtain a necessary condition for splitting T-space off a space in terms of cyclic maps, and also obtain a necessary condition for splitting co-T-spaces in terms of cocyclic maps.

• 대한수학회보
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• 제53권6호
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• pp.1617-1628
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• 2016

In this paper, we extend the results given in [3] to a nonchain ring $R_p={\mathbb{F}}_p+v{\mathbb{F}}_p+{\cdots}+v^{p-1}{\mathbb{F}}_p$, where $v^p=v$ and p is a prime. We determine the structure of the cyclic codes of arbitrary length over the ring $R_p$ and study the structure of their duals. We classify cyclic codes containing their duals over $R_p$ by giving necessary and sufficient conditions. Further, by taking advantage of the Gray map ${\pi}$ defined in [4], we give the parameters of the quantum codes of length pn over ${\mathbb{F}}_p$ which are obtained from cyclic codes over $R_p$. Finally, we illustrate the results by giving some examples.

• Journal of Computing Science and Engineering
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• 제8권4호
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• pp.187-198
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• 2014

In this paper an efficient image encryption scheme based on cyclic rotations and multiple blockwise diffusions with two chaotic maps is proposed. A Sin map is used to generate round keys for the encryption/decryption process. A Pomeau-Manneville map is used to generate chaotic values for permutation, pixel value rotation and diffusion operations. The encryption scheme is composed of three stages: permutation, pixel value rotation and diffusion. The permutation stage performs four operations on the image: row shuffling, column shuffling, cyclic rotation of all the rows and cyclic rotation of all the columns. This stage reduces the correlation significantly among neighboring pixels. The second stage performs circular rotation of pixel values twice by scanning the image horizontally and vertically. The amount of rotation is based on $M{\times}N$ chaotic values. The last stage performs the diffusion four times by scanning the image in four different ways: block of $8{\times}8$ pixels, block of $16{\times}16$ pixels, principal diagonally, and secondary diagonally. Each of the above four diffusions performs the diffusion in two directions (forwards and backwards) with two previously diffused pixels and two chaotic values. This stage makes the scheme resistant to differential attacks. The security and performance of the proposed method is analyzed systematically by using the key space, entropy, statistical, differential and performance analysis. The experimental results confirm that the proposed method is computationally efficient with high security.

• 대한수학회지
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• 제58권3호
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• pp.571-595
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• 2021

In this paper we study the algebraic structure of ℤ_pℤ_p[u]/k>-cyclic codes, where u^k = 0 and p is a prime. A ℤ_pℤ_p[u]/k>-linear code of length (r + s) is an R_k-submodule of ℤ^r_p × R^s_k with respect to a suitable scalar multiplication, where R_k = ℤ_p[u]/k>. Such a code can also be viewed as an R_k-submodule of ℤ_p[x]/r - 1> × R_k[x]/s - 1>. A new Gray map has been defined on ℤ_p[u]/k>. We have considered two cases for studying the algebraic structure of ℤ_pℤ_p[u]/k>-cyclic codes, and determined the generator polynomials and minimal spanning sets of these codes in both the cases. In the first case, we have considered (r, p) = 1 and (s, p) ≠ 1, and in the second case we consider (r, p) = 1 and (s, p) = 1. We have established the MacWilliams identity for complete weight enumerators of ℤ_pℤ_p[u]/k>-linear codes. Examples have been given to construct ℤ_pℤ_p[u]/k>-cyclic codes, through which we get codes over ℤ_p using the Gray map. Some optimal p-ary codes have been obtained in this way. An example has also been given to illustrate the use of MacWilliams identity.

• Journal of applied mathematics & informatics
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• 제34권5_6호
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• pp.397-404
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• 2016

In this work, a method is given to construct quantum codes from cyclic codes over F₄ + vF₄ which will be denoted as R throughout the paper, where v² = v and a Gray map is defined between R and where F₄ is the field with 4 elements. Some optimal quantum code parameters and others will be presented at the end of the paper.

• Journal of applied mathematics & informatics
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• 제36권5_6호
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• pp.359-368
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• 2018

In this paper, we investigate quasi-cyclic codes over the ring $R={\mathbb{F}}_2+v{\mathbb{F}}_2$, where $v^2=v$. We investigate the structure of generators for one-generator quasi-cyclic codes over R and their minimal spanning sets. Moreover, we find the rank and a lower bound on minimum distances of free quasi-cyclic codes over R. Further, we find a relationship between cyclic codes over a different ring and quasi-cyclic codes of index 2 over R.

• 한국시뮬레이션학회논문지
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• 제21권2호
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• pp.79-90
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• 2012

본 연구는 공정신호가 불균형 데이터인 경우 이상 탐지 알고리즘의 성능 개선을 위한 특징 신호 추출 기법을 제안한다. 불균형 데이터란 범주 구분 문제에서 하나의 범주의 속하는 데이터의 비율이 다른 범주의 데이터에 비해 크게 차이나 이상 탐지성능이 크게 저하되는 경우를 의미한다. 공정이 운영되는 경우 얻을 수 있는 이상 신호의 수는 정상 신호에 비해 매우 적기에 이러한 문제를 해결하여 이상 탐지 기법을 적용하는 것은 매우 중요하다. 불균형 문제 해결을 위해 SOM(Self-Organizing Map) 알고리즘을 이용하여 각 노드에 대응되는 가중치를 특징 신호로 간주하여 정상 데이터와 이상 데이터의 비율을 맞춘다. 특징 신호 데이터 집단의 이상 탐지를 위해 클래스 분류 기법인 kNN(k-Nearest Neighbor)과 SVM(Support Vector Machine)을 적용하여 이를 공정 신호 이상탐지를 위해 주로 사용하는 Hotelling's $T^2$ 관리도와 성능을 비교한다. 반도체 공정에서 발생한다고 알려진 공정 신호를 모사하여 신호 알고리즘 성능의 우수성을 검증한다.

• 대한수학회보
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• 제55권1호
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• pp.115-137
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• 2018

Let $R_{u{^2},v^2,w^2,p}$ be a finite non chain ring ${\mathbb{F}}_p[u,v,w]{\langle}u^2,\;v^2,\;w^2,\;uv-vu,\;vw-wv,\;uw-wu{\rangle}$, where p is a prime number. This ring is a part of family of Frobenius rings. In this paper, we explore the structures of cyclic codes over the ring $R_{u{^2},v^2,w^2,p}$ of arbitrary length. We obtain a unique set of generators for these codes and also characterize free cyclic codes. We show that Gray images of cyclic codes are 8-quasicyclic binary linear codes of length 8n over ${\mathbb{F}}_p$. We also determine the rank and the Hamming distance for these codes. At last, we have given some examples.

• 호남수학학술지
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• 제32권1호
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• pp.45-51
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• 2010

Let A be an abelian variety defined over a number field K and let L be a cyclic extension of K with Galois group G = <${\sigma}$> of order n. Let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and of A over L. Assume III(A/L) is finite. Let M(x) be a companion matrix of 1+x+${\cdots}$+$x^{n-1}$ and let $A^x$ be the twist of $A^{n-1}$ defined by $f^{-1}{\circ}f^{\sigma}$ = M(x) where $f:A^{n-1}{\rightarrow}A^x$ is an isomorphism defined over L. In this paper we compute [III(A/K)][III($A^x$/K)]/[III(A/L)] in terms of cohomology, where [X] is the order of an finite abelian group X.

• 대한수학회논문집
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• 제21권1호
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• pp.185-191
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• 2006

This paper observes that the induced homomorphisms on cohomology groups by a cyclic map are trivial. For a CW-complex X, we use the fact to obtain some conditions of X so that the n-th Gottlieb group $G_n(X)$ is trivial for an even positive integer n. As corollaries, for any positive integer m, we obtain $G_{2m}(S^{2m})\;=\;0\;and\;G_2(CP^m)\;=\;0$ which are due to D. H. Gottlieb and G. Lang respectively, where $S^{2m}$ is the 2m- dimensional sphere and $CP^m$ is the complex projective m-space. Moreover, we show that $G_4(HP^m)\;=\;0\;and\;G_8(II)\;=\;0,\;where\;HP^m$ is the quaternionic projective m-space for any positive integer m and II is the Cayley projective space.