• Title, Summary, Keyword: Generator Polynomials

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A NEW CLASS OF CYCLIC CODES USING ORDERED POWER PRODUCT OF POLYNOMIALS

  • Gaur, Ankita;Sharma, Bhudev
    • Journal of applied mathematics & informatics
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    • v.32 no.3_4
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    • pp.529-537
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    • 2014
  • The paper introduces a new product of polynomials defined over a field. It is a generalization of the ordinary product with inner polynomial getting non-overlapping segments obtained by multiplying with coefficients and variable with expanding powers. It has been called 'Ordered Power Product' (OPP). Considering two rings of polynomials $R_m[x]=F[x]modulox^m-1$ and $R_n[x]=F[x]modulox^n-1$, over a field F, the paper then considers the newly introduced product of the two polynomial rings. Properties and algebraic structure of the product of two rings of polynomials are studied and it is shown to be a ring. Using the new type of product of polynomials, we define a new product of two cyclic codes and devise a method of getting a cyclic code from the 'ordered power product' of two cyclic codes. Conditions for the OPP of the generators polynomials of component codes, giving a cyclic code are examined. It is shown that OPP cyclic code so obtained is more efficient than the one that can be obtained by Kronecker type of product of the same component codes.

Analysis of Shrinking Generator Using Phase Shifts (위상이동차를 이용한 수축 생성기의 분석)

  • Hwang, Yoon-Hee;Cho, Sung-Jin;Choi, Un-Sook
    • Journal of the Korea Institute of Information and Communication Engineering
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    • v.14 no.11
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    • pp.2507-2513
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    • 2010
  • In this paper, we show that the shrinking generator with two LFSR whose characteristic polynomials are primitive is an interleaving generator and analyze phase shifts in shrunken sequence. Also for a given intercepted sequence of shrunken sequence, we propose. the method of reconstructing some deterministic bits of the shrunken sequence using phase shifts.

A Study on the High Speed Curve Generator Using 1-Dimensional Systolic Array Processor (1차원 시스톨릭 어레이 프로세서를 이용한 고속 곡선 발생기에 관한 연구)

  • 김용성;조원경
    • Journal of the Korean Institute of Telematics and Electronics B
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    • v.31B no.5
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    • pp.1-11
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    • 1994
  • In computer graphics since objects atre constructed by lines and curves, the high-speed curve generator is indispensible for computer aided design and simulatation. Since the functions of graphic generation can be represented as a series of matrix operations, in this paper, two kind of the high-speed Bezier curve generator that uses matrix equation and a recursive relation for Bezier polynomials are designed. And B-spline curve generator is designed using interdependence of B-spline blending functions. As the result of the comparison of designed curve generator and reference [5], [6] in the operation time and number of operators, the curve generator with 1-dimensional systolic array processor for matrix vector operation that uses matrix equation for Bezier curve is more effective.

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A Study on a Binary Random Sequence Generator with Two Characteristic Polynomials (두개의 특성 다항식으로 구성된 이진 난수열 발생기에 관한 연구)

  • 김대엽;주학수;임종인
    • Journal of the Korea Institute of Information Security & Cryptology
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    • v.12 no.3
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    • pp.77-85
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    • 2002
  • A Research of binary random sequence generator that uses a linear shift register had been studied since the 1970s. These generators were used in stream cipher. In general, the binary random sequence generator consists of linear shift registers that generate sequences of maximum period and a nonlinear filter function or a nonlinear combination function to generate a sequence of high linear complexity. Therefore, To generate a sequence that have long period as well as high linear complexity becomes an important factor to estimate safety of stream cipher. Usually, the maximum period of the sequence generated by a linear feedback shift register with L resistors is less than or equal to $2^L$-1. In this paper, we propose new binary random sequence generator that consist of L registers and 2 sub-characteristic polynomials. According to an initial state vector, the least period of the sequence generated by the proposed generator is equal to or ions than it of the sequence created by the general linear feedback shift register, and its linear complexity is increased too.

On the Construction of the 90/150 State Transition Matrix Corresponding to the Trinomial x2n-1 + x + 1 (3항 다항식 x2n-1 + x + 1에 대응하는 90/150 상태전이행렬의 구성)

  • Kim, Han-Doo;Cho, Sung-Jin;Choi, Un-Sook
    • The Journal of the Korea institute of electronic communication sciences
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    • v.13 no.2
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    • pp.383-390
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    • 2018
  • Since cellular automata(CA) is superior to LFSR in randomness, it is applied as an alternative of LFSR in various fields. However, constructing CA corresponding to a given polynomial is more difficult than LFSR. Cattell et al. and Cho et al. showed that irreducible polynomials are CA-polynomials. And Cho et al. and Sabater et al. gave a synthesis method of 90/150 CA corresponding to the power of an irreducible polynomial, which is applicable as a shrinking generator. Swan characterizes the parity of the number of irreducible factors of a trinomial over the finite field GF(2). These polynomials are of practical importance when implementing finite field extensions. In this paper, we show that the trinomial $x^{2^n-1}+X+1$ ($n{\geq}2$) are CA-polynomials. Also the trinomial $x^{2^a(2^n-1)}+x^{2^a}+1$ ($n{\geq}2$, $a{\geq}0$) are CA-polynomials.

REMARK ON THE MEAN VALUE OF L(½, χ) IN THE HYPERELLIPTIC ENSEMBLE

  • Jung, Hwanyup
    • Journal of the Chungcheong Mathematical Society
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    • v.27 no.1
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    • pp.9-16
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    • 2014
  • Let $\mathbb{A}=\mathbb{F}_q[T]$ be a polynomial ring over $\mathbb{F}_q$. In this paper we determine an asymptotic mean value of quadratic Dirich-let L-functions L(s, ${\chi}_{{\gamma}D}$) at the central point s=$\frac{1}{2}$, where D runs over all monic square-free polynomials of even degree in $\mathbb{A}$ and ${\gamma}$ is a generator of $\mathbb{F}_q^*$.

Design of A Turbo-code Decoder for Speech Transmission in IMT-2000 (IMT-2000에서 음성 전송을 위한 터보 코드 복호기 설계)

  • 강태환;박성모
    • Proceedings of the IEEK Conference
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    • pp.273-276
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    • 2000
  • Recently, Turbo code has been considered for channel coding in IMT-2000(International Mobile Telecommunication-2000) system, because it offers better error correcting capability than the traditional convolution/viterbi coding . In this paper, a turbo code decoder for speech transmission in IMT-2000 system with frame size 192 bits, constrait length K=3, generator polynomials G(5,7) and code rate R=1/3 is designed using SOVA(Soft Output Viterbi Algorithm) and block interleaver

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NONCOMMUTATIVE CONTINUOUS FUNCTIONS

  • Don, Hadwin;Llolsten, Kaonga;Ben, Mathes
    • Journal of the Korean Mathematical Society
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    • v.40 no.5
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    • pp.789-830
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    • 2003
  • By forming completions of families of noncommutative polynomials, we define a notion of noncommutative continuous function and locally bounded Borel function that give a noncommutative analogue of the functional calculus for elements of commutative $C^{*}$-algebras and von Neumann algebras. These notions give a precise meaning to $C^{*}$-algebras defined by generator and relations and we show how they relate to many parts of operator and operator algebra theory.

A Study on Design of High-Speed Parallel Multiplier over GF(2m) using VCG (VCG를 사용한 GF(2m)상의 고속병렬 승산기 설계에 관한 연구)

  • Seong, Hyeon-Kyeong
    • Journal of the Korea Institute of Information and Communication Engineering
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    • v.14 no.3
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    • pp.628-636
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    • 2010
  • In this paper, we present a new type high speed parallel multiplier for performing the multiplication of two polynomials using standard basis in the finite fields GF($2^m$). Prior to construct the multiplier circuits, we design the basic cell of vector code generator(VCG) to perform the parallel multiplication of a multiplicand polynomial with a irreducible polynomial and design the partial product result cell(PPC) to generate the result of bit-parallel multiplication with one coefficient of a multiplicative polynomial with VCG circuits. The presented multiplier performs high speed parallel multiplication to connect PPC with VCG. The basic cell of VCG and PPC consists of one AND gate and one XOR gate respectively. Extending this process, we show the design of the generalized circuits for degree m and a simple example of constructing the multiplier circuit over finite fields GF($2^4$). Also, the presented multiplier is simulated by PSpice. The multiplier presented in this paper uses the VCGs and PPCS repeatedly, and is easy to extend the multiplication of two polynomials in the finite fields with very large degree m, and is suitable to VLSL.