• Title/Summary/Keyword: generator polynomial

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ON THE RATIONAL COHOMOLOGY OF MAPPING SPACES AND THEIR REALIZATION PROBLEM

  • Abdelhadi Zaim
    • Communications of the Korean Mathematical Society
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    • v.38 no.4
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    • pp.1309-1320
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    • 2023
  • Let f : X → Y be a map between simply connected CW-complexes of finite type with X finite. In this paper, we prove that the rational cohomology of mapping spaces map(X, Y ; f) contains a polynomial algebra over a generator of degree N, where N = max{i, πi(Y)⊗ℚ ≠ 0} is an even number. Moreover, we are interested in determining the rational homotopy type of map(𝕊n, ℂPm; f) and we deduce its rational cohomology as a consequence. The paper ends with a brief discussion about the realization problem of mapping spaces.

On Fast M-Gold Hadamard Sequence Transform (고속 M-Gold-Hadamard 시퀀스 트랜스폼)

  • Lee, Mi-Sung;Lee, Moon-Ho;Park, Ju-Yong
    • Journal of the Institute of Electronics Engineers of Korea TC
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    • v.47 no.7
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    • pp.93-101
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    • 2010
  • In this paper we generate Gold-sequence by using M-sequence which is made by two primitive polynomial of GF(2). Generally M-sequence is generated by linear feedback shift register code generator. Here we show that this matrix of appropriate permutation has Hadamard matrix property. This matrix proves that Gold-sequence through two M-sequence and additive matrix of one column has one of major properties of Hadamard matrix, orthogonal. and this matrix show another property that multiplication with one matrix and transpose matrix of this matrix have the result of unit matrix. Also M-sequence which is made by linear feedback shift register gets Hadamard matrix property mentioned above by adding matrices of one column and one row. And high-speed conversion is possible through L-matrix and the S-matrix.

An Improved Reconstruction Algorithm of Convolutional Codes Based on Channel Error Rate Estimation (채널 오류율 추정에 기반을 둔 길쌈부호의 개선된 재구성 알고리즘)

  • Seong, Jinwoo;Chung, Habong
    • The Journal of Korean Institute of Communications and Information Sciences
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    • v.42 no.5
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    • pp.951-958
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    • 2017
  • In an attack context, the adversary wants to retrieve the message from the intercepted noisy bit stream without any prior knowledge of the channel codes used. The process of finding out the code parameters such as code length, dimension, and generator, for this purpose, is called the blind recognition of channel codes or the reconstruction of channel codes. In this paper, we suggest an improved algorithm of the blind recovery of rate k/n convolutional encoders in a noisy environment. The suggested algorithm improves the existing algorithm by Marazin, et. al. by evaluating the threshold value through the estimation of the channel error probability of the BSC. By applying the soft decision method by Shaojing, et. al., we considerably enhance the success rate of the channel reconstruction.

Analysis of Code Sequence Generating Algorithm and Its Implementation based on Normal Bases for Encryption (암호화를 위한 정규기저 기반 부호계열 발생 알고리즘 분석 및 발생기 구성)

  • Lee, Jeong-Jae
    • Journal of the Institute of Convergence Signal Processing
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    • v.15 no.2
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    • pp.48-54
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    • 2014
  • For the element ${\alpha}{\in}GF(p^n)$, two kinds of bases are known. One is a conventional polynomial basis of the form $\{1,{\alpha},{\alpha}^2,{\cdots},{\alpha}^{n-1}\}$, and the other is a normal basis of the form $\{{\alpha},{\alpha}^p,{\alpha}^{p^2},{\cdots},{\alpha}^{p^{n-1}}\}$. In this paper we consider the method of generating normal bases which construct the finite field $GF(p^n)$, as an n-dimensional extension of the finite field GF(p). And we analyze the code sequence generating algorithm and derive the implementation functions of code sequence generator based on the normal bases. We find the normal polynomials of degrees, n=5 and n=7, which can generate normal bases respectively, design, and construct the code sequence generators based on these normal bases. Finally, we produce two code sequence groups(n=5, n=7) by using Simulink, and analyze the characteristics of the autocorrelation function, $R_{i,i}(\tau)$, and crosscorrelation function, $R_{i,j}(\tau)$, $i{\neq}j$ between two different code sequences. Based on these results, we confirm that the analysis of generating algorithms and the design and implementation of the code sequence generators based on normal bases are correct.

Stereo Vision based on Planar Algebraic Curves (평면대수곡선을 기반으로 한 스테레오 비젼)

  • Ahn, Min-Ho;Lee, Chung-Nim
    • Journal of KIISE:Software and Applications
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    • v.27 no.1
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    • pp.50-61
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    • 2000
  • Recently the stereo vision based on conics has received much attention by many authors. Conics have many features such as their matrix expression, efficient correspondence checking, abundance of conical shapes in real world. Extensions to higher algebraic curves met with limited success. Although irreducible algebraic curves are rather rare in the real world, lines and conics are abundant whose products provide good examples of higher algebraic curves. We consider plane algebraic curves of an arbitrary degree $n{\geq}2$ with a fully calibrated stereo system. We present closed form solutions to both correspondence and reconstruction problems. Let $f_1,\;f_2,\;{\pi}$ be image curves and plane and $VC_P(g)$ the cone with generator (plane) curve g and vertex P. Then the relation $VC_{O1}(f_1)\;=\;VC_{O1}(VC_{O2}(f_2)\;∩\;{\pi})$ gives polynomial equations in the coefficient $d_1,\;d_2,\;d_3$ of the plane ${\pi}$. After some manipulations, we get an extremely simple polynomial equation in a single variable whose unique real positive root plays the key role. It is then followed by evaluating $O(n^2)$ polynomials of a single variable at the root. It is in contrast to the past works which usually involve a simultaneous system of multivariate polynomial equations. We checked our algorithm using synthetic as well as real world images.

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Sufficient Conditions for the Existence of an (n, 1) Mother Code and Its Puncturing Pattern to Generating a Given Convolutional Code (임의의 생성다항식 행렬을 갖는 길쌈부호도 (n, 1) 마더부호의 천공으로 생성 가능한가?)

  • Chung, Habong;Seong, Jinwoo
    • The Journal of Korean Institute of Communications and Information Sciences
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    • v.41 no.4
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    • pp.379-386
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    • 2016
  • Puncturing is the most common way of increasing the rate of convolutional codes. The puncturing process is done to the original code called the mother code by a specific puncturing pattern. In this article, we investigate into the question whether any convolutional code is obtainable by puncturing some (n, 1) mother codes. We present two sufficient conditions for the mother code and the puncturing pattern to satisfy in order that the punctured code is equivalent to the given (N, K) convolutional code.

On the Existence of the (2,1) Mother Code of (n,n-1) Convolutional Code ((n,n-1) 길쌈부호에 대한 (2,1) 마더부호의 존재)

  • Jang, Hwan-Seok;Chung, Ha-Bong;Seong, Jin-Woo
    • The Journal of Korean Institute of Communications and Information Sciences
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    • v.39A no.4
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    • pp.165-171
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    • 2014
  • The rate of the channel code can be controlled by various methods. Puncturing is one of the methods of increasing the code rate, and the original code before puncturing is called the mother code. Any (n,k) convolutional code is obtainable by puncturing some mother codes, and the process of finding the mother code is necessary for designing the optimum channel decoder. In this paper, we proved that any (n,n-1) convolutional code has (2,1) mother codes regardless of the puncturing pattern and showed that they must be equivalent.

Comparison of Parallel CRC Verification Algorithms for ATM Cell Delineation (ATM 셀 경계식별을 위한 병렬 CRC 검증 알고리즘의 비교)

  • 최윤희;송상섭
    • The Journal of Korean Institute of Communications and Information Sciences
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    • v.18 no.11
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    • pp.1655-1662
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    • 1993
  • In this paper we discuss three algorithms-Direct, Successive, and Recursive-on parallel CRC(Cyclic Redundancy Check) verification. The algorithms are derived by combining the byte-syndromes precomputed from the generator polynomial. These algorithms are compared in terms of the amount of hardware and the speed of operation. Since the algorithms can be generalized easily, we took the ATM cell delineation example for easier description. As an application of the algorithm Recursive, an ATM cell delineation module suitable for STM-1 transmission has been successfully realized through commercially available field programmable gate arrays.

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Synthesis of Symmetric 1-D 5-neighborhood CA using Krylov Matrix (Krylov 행렬을 이용한 대칭 1차원 5-이웃 CA의 합성)

  • Cho, Sung-Jin;Kim, Han-Doo;Choi, Un-Sook;Kang, Sung-Won
    • The Journal of the Korea institute of electronic communication sciences
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    • v.15 no.6
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    • pp.1105-1112
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    • 2020
  • One-dimensional 3-neighborhood Cellular Automata (CA)-based pseudo-random number generators are widely applied in generating test patterns to evaluate system performance and generating key sequence generators in cryptographic systems. In this paper, in order to design a CA-based key sequence generator that can generate more complex and confusing sequences, we study a one-dimensional symmetric 5-neighborhood CA that expands to five neighbors affecting the state transition of each cell. In particular, we propose an n-cell one-dimensional symmetric 5-neighborhood CA synthesis algorithm using the algebraic method that uses the Krylov matrix and the one-dimensional 90/150 CA synthesis algorithm proposed by Cho et al. [6].

Generalization of Galois Linear Feedback Register (갈로이 선형 궤환 레지스터의 일반화)

  • Park Chang-Soo;Cho Gyeong-Yeon
    • Journal of the Institute of Electronics Engineers of Korea CI
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    • v.43 no.1 s.307
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    • pp.1-8
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    • 2006
  • This thesis proposes Arithmetic Shift Register(ASR) which can be used as pseudo random number generator. Arithmetic Shift. Register is defined as progression that multiplies random number D , not 0 or 1 at initial value which is not 0, and it is represented as ASR-D in this thesis. Irreducible polynomial that t which makes $'D^k=1'$ satisfies uniquely as $'t=2^n-1'$ over. $GF(2^n)$ is the characteristic polynomial of ASR-D , and the cycle of Arithmetic Shift Register has maximum cycle as $'2^n-1'$. Galois Linear Feedback Shift Register corresponds to ASR-2-1. Therefore, Arithmetic Shift Register proposed in this thesis generalizes Galois Linear Feedback Shift Register. Linear complexity of ASR-D over$GF(2^n)$ is $'n{\leq}LC{\leq}\frac{n^2+n}{2}'$ and in comparison with existing Linear Feedback Shift Register stability is high. The Software embodiment of arithmetic shift register proposed in this thesis is efficient than that of existing Linear Shift Register and hardware complexity is equal. Arithmetic shift register proposed in this thesis can be used widely in various fields such as cipher, error correcting codes, Monte Carlo integral, and data communication etc along with existing linear shift register.