The Experts below are selected from a list of 2046 Experts worldwide ranked by ideXlab platform
Hajime Matsui - One of the best experts on this subject based on the ideXlab platform.
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Quasi-Cyclic Codes Via Unfolded Cyclic Codes and Their Reversibility
IEEE Access, 2020Co-Authors: Ramy Taki Eldin, Hajime MatsuiAbstract:The finite field $\mathbb {F}_{q^\ell }$ of $q^\ell $ elements contains $\mathbb {F}_{q}$ as a subfield. If $\theta \in \mathbb {F}_{q^\ell }$ is of degree $\ell $ over $\mathbb {F}_{q}$ , it can be used to unfold elements of $\mathbb {F}_{q^\ell }$ to vectors in $\mathbb {F}_{q}^\ell $ . We apply the unfolding to the coordinates of all codewords of a cyclic code $\mathcal {C}$ over $\mathbb {F}_{q^\ell }$ of length $n$ . This generates a quasi-cyclic code $\mathcal {Q}$ over $\mathbb {F}_{q}$ of length $n\ell $ and index $\ell $ . We focus on the class of quasi-cyclic codes resulting from the unfolding of cyclic codes. Given a Generator Polynomial $g(x)$ of a cyclic code $\mathcal {C}$ , we present a formula for a Generator Polynomial matrix for the unfolded code $\mathcal {Q}$ . On the other hand, for any quasi-cyclic code $\mathcal {Q}$ with a reduced Generator Polynomial matrix $G$ , we provide a necessary and sufficient condition on $G$ that determines whether or not the code $\mathcal {Q}$ can be represented as the unfolding of a cyclic code. Furthermore, as an application, we discuss the reversibility of the class of quasi-cyclic codes resulting from unfolding of cyclic codes. Specifically, we provide a necessary and sufficient condition on the defining set $\mathcal {T}$ of the cyclic code $\mathcal {C}$ that ensures the reversibility of the unfolded code. Numerical examples are used to illustrate theoretical results. Some of these examples show that quasi-cyclic codes reversibility does not necessarily require a self-reciprocal Generator Polynomial for the cyclic code. Since reversibility is essential in constructing DNA codes, some DNA codes are designed as examples.
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Quasi-Cyclic Codes Via Unfolded Cyclic Codes and Their Reversibility
IEEE Access, 2019Co-Authors: Ramy Taki Eldin, Hajime MatsuiAbstract:The finite field Fq(ℓ) of qℓ elements contains Fq as a subfield. If θ ∈ Fq(ℓ) is of degree ℓ over Fq, it can be used to unfold elements of Fq(ℓ) to vectors in Fℓq. We apply the unfolding to the coordinates of all codewords of a cyclic code C over Fq(ℓ) of length n. This generates a quasi-cyclic code Q over Fq of length nℓ and index ℓ. We focus on the class of quasi-cyclic codes resulting from the unfolding of cyclic codes. Given a Generator Polynomial g(x) of a cyclic code C, we present a formula for a Generator Polynomial matrix for the unfolded code Q. On the other hand, for any quasi-cyclic code Q with a reduced Generator Polynomial matrix G, we provide a necessary and sufficient condition on G that determines whether or not the code Q can be represented as the unfolding of a cyclic code. Furthermore, as an application, we discuss the reversibility of the class of quasi-cyclic codes resulting from unfolding of cyclic codes. Specifically, we provide a necessary and sufficient condition on the defining set T of the cyclic code C that ensures the reversibility of the unfolded code. Numerical examples are used to illustrate theoretical results. Some of these examples show that quasi-cyclic codes reversibility does not necessarily require a self-reciprocal Generator Polynomial for the cyclic code. Since reversibility is essential in constructing DNA codes, some DNA codes are designed as examples.
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on Generator and parity check Polynomial matrices of generalized quasi cyclic codes
Finite Fields and Their Applications, 2015Co-Authors: Hajime MatsuiAbstract:Generalized quasi-cyclic (GQC) codes have been investigated as well as quasi-cyclic (QC) codes, e.g., on the construction of efficient low-density parity-check codes. While QC codes have the same length of cyclic intervals, GQC codes have different lengths of cyclic intervals. Similarly to QC codes, each GQC code can be described by an upper triangular Generator Polynomial matrix, from which the systematic encoder is constructed. In this paper, a complete theory of Generator Polynomial matrices of GQC codes, including a relation formula between Generator Polynomial matrices and parity-check Polynomial matrices through their equations, is provided. This relation generalizes those of cyclic codes and QC codes. While the previous researches on GQC codes are mainly concerned with 1-Generator case or linear algebraic approach, our argument covers the general case and shows the complete analogy of QC case. We do not use Grobner basis theory explicitly in order that all arguments of this paper are self-contained. Numerical examples are attached to the dual procedure that extracts one from each other. Finally, we provide an efficient algorithm which calculates all Generator Polynomial matrices with given cyclic intervals.
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On Generator Polynomial matrices of generalized pseudo-cyclic codes
2014 International Symposium on Information Theory and its Applications, 2014Co-Authors: Hajime MatsuiAbstract:In this paper, generalized pseudo-cyclic (GPC) codes are proposed and their basic properties are investigated. Generator Polynomial matrices of GPC codes are constructively defined and thereby a dimension formula for GPC codes is provided. While a pseudo-cyclic code is equal to an ideal of the ring which consists of Polynomials modulo a fixed Polynomial, a GPC code is equal to a submodule of the direct sum of the rings which consist of Polynomials modulo fixed Polynomials. In the theory of cyclic codes and PC codes, Euclidean division algorithm for Polynomials is essential; its generalization for Polynomial vectors is obtained and enables us with Generator Polynomial matrices to encode GPC codes fast and systematically.
Yu Wang - One of the best experts on this subject based on the ideXlab platform.
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Some Results on Cyclic Codes Over ${F}_{2}+v{F}_{2}$
IEEE Transactions on Information Theory, 2010Co-Authors: Yu WangAbstract:In this paper, we investigate the structure and properties of cyclic codes over the ring F 2+vF 2 . We first study the relationship between cyclic codes over F 2+vF 2 and binary cyclic codes. Then we prove that cyclic codes over the ring are principally generated, and give the Generator Polynomial of cyclic codes over the ring. Finally, we obtain the unique idempotent Generators for cyclic codes of odd length and determine the number of cyclic codes for a given length n over F 2+vF 2.
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Cyclic codes over F2 + vF2
2009 IEEE International Symposium on Information Theory, 2009Co-Authors: Yu WangAbstract:In this paper, we investigate the structure and properties of cyclic codes over the ring F2 + vF2. We first study the relationship between cyclic codes over F2 + vF2 and binary cyclic codes. Then we prove that cyclic codes over the ring are principally generated, and give the Generator Polynomial of cyclic codes over the ring. Finally, we obtain the unique idempotent Generators for cyclic codes of odd length over F2 + vF2.
Arash Reyhani-masoleh - One of the best experts on this subject based on the ideXlab platform.
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High-Speed CRC Computations Using
2020Co-Authors: Christopher Kennedy, Arash Reyhani-masolehAbstract:Previously, a state-space similarity transform was proposed to reduce the feedback loop complexity of a parallel Cyclic Redundancy Check architecture and enable retiming. This paper investigates the open research question concerning the impact of varying the vector used to construct the transformation matrix. We perform exhaustive searches of the vector space for frequently referenced Generator Polynomials when the input size is equal to the degree of the Generator Polynomial. The set of vectors which yield minimal hardware state-space representations is obtained. Then, application-specific integrated circuit (ASIC) experiments are performed. The ASIC implementation results for the minimized state spaces demonstrate improvement in both area and timing as compared to the original ones. Finally, it is concluded that the vectors obtained for a fixed Generator Polynomial are also good choices for other input sizes.
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High-speed CRC computations using improved state-space transformations
2009 IEEE International Conference on Electro Information Technology, 2009Co-Authors: Christopher Kennedy, Arash Reyhani-masolehAbstract:Previously, a state-space similarity transform was proposed to reduce the feedback loop complexity of a parallel Cyclic Redundancy Check architecture and enable retiming. This paper investigates the open research question concerning the impact of varying the vector used to construct the transformation matrix. We perform exhaustive searches of the vector space for frequently referenced Generator Polynomials when the input size is equal to the degree of the Generator Polynomial. The set of vectors which yield minimal hardware state-space representations is obtained. Then, application-specific integrated circuit (ASIC) experiments are performed. The ASIC implementation results for the minimized state spaces demonstrate improvement in both area and timing as compared to the original ones. Finally, it is concluded that the vectors obtained for a fixed Generator Polynomial are also good choices for other input sizes.
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EIT - High-speed CRC computations using improved state-space transformations
2009 IEEE International Conference on Electro Information Technology, 2009Co-Authors: Christopher Kennedy, Arash Reyhani-masolehAbstract:Previously, a state-space similarity transform was proposed to reduce the feedback loop complexity of a parallel Cyclic Redundancy Check architecture and enable retiming. This paper investigates the open research question concerning the impact of varying the vector used to construct the transformation matrix. We perform exhaustive searches of the vector space for frequently referenced Generator Polynomials when the input size is equal to the degree of the Generator Polynomial. The set of vectors which yield minimal hardware state-space representations is obtained. Then, application-specific integrated circuit (ASIC) experiments are performed. The ASIC implementation results for the minimized state spaces demonstrate improvement in both area and timing as compared to the original ones. Finally, it is concluded that the vectors obtained for a fixed Generator Polynomial are also good choices for other input sizes.
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High-speed parallel CRC circuits
2008 42nd Asilomar Conference on Signals Systems and Computers, 2008Co-Authors: C. Kennedy, Arash Reyhani-masolehAbstract:In this paper, we develop a matrix-based formulation for the Cyclic Redundancy Check (CRC) computation that is derived from its Polynomial-based definition. Then, using this formulation, we propose a parallel CRC computation structure with optimizations specific to the case when the degree of parallelism is greater than the degree of the Generator Polynomial. Afterward, through extensive simulations we obtain the optimum degrees of parallelism in terms of their critical path delays for some common Generator Polynomials. We also show that the time-area product follows the critical path delay plot.
Suhas Kulkarni - One of the best experts on this subject based on the ideXlab platform.
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impact of Generator Polynomial on performance of map turbo decoder in awgn channel
Wireless Communication, 2011Co-Authors: S V Viraktamath, Attimarad Attimarad, Praveen M Purohit, Shweta S Kulkarni, Suhas KulkarniAbstract:Turbo coding is the most commonly used error correcting scheme in wireless systems resulting in maximum Coding gain. The MAP algorithm is a basic algorithm for turbo decoding. In this paper, authors analyze the performance of turbo MAP algorithm in terms of bit error rate (BER), considering different parameters like signal to noise ratio (SNR), Generator Polynomials and length of Generator Polynomial on Additive White Gaussian Noise (AWGN) channel. Simulation of Turbo encoder and Turbo MAP decoder is done. BER and processing time are computed for a range of SNR,considering different Generator Polynomials of same length and also for Generator Polynomials of different length, keeping the input constant. Simulation results show that BER for all the Generator Polynomial of same length is not same for a given SNR. Same is the case with processing time. BER for a given SNR decreases as the length of Generator Polynomial is increased whereas processing time increases with increase in Generator Polynomial length.
Ramy Taki Eldin - One of the best experts on this subject based on the ideXlab platform.
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Quasi-Cyclic Codes Via Unfolded Cyclic Codes and Their Reversibility
IEEE Access, 2020Co-Authors: Ramy Taki Eldin, Hajime MatsuiAbstract:The finite field $\mathbb {F}_{q^\ell }$ of $q^\ell $ elements contains $\mathbb {F}_{q}$ as a subfield. If $\theta \in \mathbb {F}_{q^\ell }$ is of degree $\ell $ over $\mathbb {F}_{q}$ , it can be used to unfold elements of $\mathbb {F}_{q^\ell }$ to vectors in $\mathbb {F}_{q}^\ell $ . We apply the unfolding to the coordinates of all codewords of a cyclic code $\mathcal {C}$ over $\mathbb {F}_{q^\ell }$ of length $n$ . This generates a quasi-cyclic code $\mathcal {Q}$ over $\mathbb {F}_{q}$ of length $n\ell $ and index $\ell $ . We focus on the class of quasi-cyclic codes resulting from the unfolding of cyclic codes. Given a Generator Polynomial $g(x)$ of a cyclic code $\mathcal {C}$ , we present a formula for a Generator Polynomial matrix for the unfolded code $\mathcal {Q}$ . On the other hand, for any quasi-cyclic code $\mathcal {Q}$ with a reduced Generator Polynomial matrix $G$ , we provide a necessary and sufficient condition on $G$ that determines whether or not the code $\mathcal {Q}$ can be represented as the unfolding of a cyclic code. Furthermore, as an application, we discuss the reversibility of the class of quasi-cyclic codes resulting from unfolding of cyclic codes. Specifically, we provide a necessary and sufficient condition on the defining set $\mathcal {T}$ of the cyclic code $\mathcal {C}$ that ensures the reversibility of the unfolded code. Numerical examples are used to illustrate theoretical results. Some of these examples show that quasi-cyclic codes reversibility does not necessarily require a self-reciprocal Generator Polynomial for the cyclic code. Since reversibility is essential in constructing DNA codes, some DNA codes are designed as examples.
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Quasi-Cyclic Codes Via Unfolded Cyclic Codes and Their Reversibility
IEEE Access, 2019Co-Authors: Ramy Taki Eldin, Hajime MatsuiAbstract:The finite field Fq(ℓ) of qℓ elements contains Fq as a subfield. If θ ∈ Fq(ℓ) is of degree ℓ over Fq, it can be used to unfold elements of Fq(ℓ) to vectors in Fℓq. We apply the unfolding to the coordinates of all codewords of a cyclic code C over Fq(ℓ) of length n. This generates a quasi-cyclic code Q over Fq of length nℓ and index ℓ. We focus on the class of quasi-cyclic codes resulting from the unfolding of cyclic codes. Given a Generator Polynomial g(x) of a cyclic code C, we present a formula for a Generator Polynomial matrix for the unfolded code Q. On the other hand, for any quasi-cyclic code Q with a reduced Generator Polynomial matrix G, we provide a necessary and sufficient condition on G that determines whether or not the code Q can be represented as the unfolding of a cyclic code. Furthermore, as an application, we discuss the reversibility of the class of quasi-cyclic codes resulting from unfolding of cyclic codes. Specifically, we provide a necessary and sufficient condition on the defining set T of the cyclic code C that ensures the reversibility of the unfolded code. Numerical examples are used to illustrate theoretical results. Some of these examples show that quasi-cyclic codes reversibility does not necessarily require a self-reciprocal Generator Polynomial for the cyclic code. Since reversibility is essential in constructing DNA codes, some DNA codes are designed as examples.