Distributed storage systems are composed by many unreliable storage nodes over a network. A data file is redundantly stored in multiple storage nodes to provide high reliability. Recently erasure codes with Maximum Di...
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ISBN:
(纸本)9781509029914
Distributed storage systems are composed by many unreliable storage nodes over a network. A data file is redundantly stored in multiple storage nodes to provide high reliability. Recently erasure codes with Maximum Distance Separable (MDS) property are gradually employed in distributed storage systems to reduce the cost of reliably storing large amounts of data. Regenerating codes are a class of erasure codes which can achieve the optimal trade-off between the storage capacity and the bandwidth needed to repair a failed node. However, one of the critical drawbacks of existing MDS erasure codes in general is the high coding and repair complexities, since the coding and repair processes involve expensive multiplication operations in a finite field. binary Addition and Shift Implementable cyclic-convolutional (BASIC) codes, which is a coding framework of linear codes with a binary cyclic code as the alphabet, were proposed recently with lower computational complexity by replacing a finite field multiplication by a cyclic-shift operation. This paper provides an overview of the existing results of BASIC codes, and proposes several interesting open problems about BASIC codes.
Distributed storage systems are composed by many unreliable storage nodes over a network. A data file is redundantly stored in multiple storage nodes to provide high reliability. Recently erasure codes with Maximum Di...
详细信息
Distributed storage systems are composed by many unreliable storage nodes over a network. A data file is redundantly stored in multiple storage nodes to provide high reliability. Recently erasure codes with Maximum Distance Separable (MDS) property are gradually employed in distributed storage systems to reduce the cost of reliably storing large amounts of data. Regenerating codes are a class of erasure codes which can achieve the optimal trade-off between the storage capacity and the bandwidth needed to repair a failed node. However, one of the critical drawbacks of existing MDS erasure codes in general is the high coding and repair complexities, since the coding and repair processes involve expensive multiplication operations in a finite field. binary Addition and Shift Implementable cyclicconvolutional (BASIC) codes, which is a coding framework of linear codes with a binary cyclic code as the alphabet, were proposed recently with lower computational complexity by replacing a finite field multiplication by a cyclic-shift operation. This paper provides an overview of the existing results of BASIC codes, and proposes several interesting open problems about BASIC codes.
cycliccodes form an important class of codes. They have very interesting algebraic structure. Furthermore, they are equivalent to many important codes, such as binary Hamming codes, Golay codes and BCH codes. Minimal...
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cycliccodes form an important class of codes. They have very interesting algebraic structure. Furthermore, they are equivalent to many important codes, such as binary Hamming codes, Golay codes and BCH codes. Minimal codewords in linear codes are widely used in constructing decoding algorithms and studying linear secret sharing scheme. In this paper, we show that in the binary cyclic code all of the codewords are minimal, except 0 and 1. Then, we obtain a result about the number of minimal codewords in the binary cyclic codes.
Currently, there has been an increasing demand for operational and trustworthy digital data transmission and storage systems. This demand has been augmented by the appearance of large-scale, high-speed data networks f...
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Currently, there has been an increasing demand for operational and trustworthy digital data transmission and storage systems. This demand has been augmented by the appearance of large-scale, high-speed data networks for the exchange, processing and storage of digital information in the different spheres. In this paper, we explore a way to achieve this goal. For given positive integers , we establish that corresponding to a binary cyclic code , there is a binary cyclic code , where is a nonnegative integer, which plays a role in enhancing code rate and error correction capability. In the given scheme, the new code is in fact responsible to carry data transmitted by . Consequently, a codeword of the code can be encoded by the generator matrix of and therefore this arrangement for transferring data offers a safe and swift mode.
A new bound on the distance of binary cyclic codes is proposed. The approach is based on the representation of a subset of the roots of the generator polynomial by a rational function. A new bound on the minimum dista...
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ISBN:
(纸本)9781457705953
A new bound on the distance of binary cyclic codes is proposed. The approach is based on the representation of a subset of the roots of the generator polynomial by a rational function. A new bound on the minimum distance is proven and several classes of binary cyclic codes are identified. For some classes of codes, this bound is better than the known bounds (e. g. BCH or Hartmann-Tzeng bound). Furthermore, a quadratic-time decoding algorithm up to this new bound is developed.
Optical encoder is the important sensor used for length and angle measurement. A new type optical encoder: virtual absolute encoder has inherited the advantage of the two old traditional ones. The principle of virtual...
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Optical encoder is the important sensor used for length and angle measurement. A new type optical encoder: virtual absolute encoder has inherited the advantage of the two old traditional ones. The principle of virtual absolute encoder has been introduced. A practical coding algorithm was designed according to the characters of the slit disk. The code design of indexing track highest to 17 bits comes true with the aid of computer aided design (CAD). A method to decode was also developed, which can convert the binary cyclic code into binary natural code, and this method furnishes the further engineering practice with the theoretical foundation.
Golomb and Gong ([8] and [9]) considered binary sequences with the trinomial property. In this correspondence me shall show that the sets of those sequences are (quite trivially) closely connected with binary-cyclic c...
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Golomb and Gong ([8] and [9]) considered binary sequences with the trinomial property. In this correspondence me shall show that the sets of those sequences are (quite trivially) closely connected with binary-cycliccodes with codewords of weight three (which were already studied in [4] and [5]). This approach gives us another way to deal with trinomial property problems. After disproving one conjecture formulated by Golomb and Gong in [9], we exhibit an infinite class of sequences which do not have the trinomial property, corresponding to binary cyclic codes of length 2(m) - 1 with minimum distance exactly four.
We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are (2(d) - 1, 2(d-1) - 1, 2(d-2) - 1) cyclic difference sets in the multiplicative g...
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We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are (2(d) - 1, 2(d-1) - 1, 2(d-2) - 1) cyclic difference sets in the multiplicative group of the finite field F-2d of 2(d) elements, with d greater than or equal to 2. We show that, except for a few cases with small d, these difference sets are all pairwise inequivalent. This is accomplished in part by examining their 2-ranks. The 2-ranks of all of these difference sets were previously known, except for those connected with the Segre and Glynn hyperovals. We determine the 2-ranks of the difference sets arising from the Segre and Glynn hyperovals, in the following way. Stickelberger's theorem for Gauss sums is used to reduce the computation of these 2-ranks to a problem of counting certain cyclicbinary strings of length d. This counting problem is then solved combinatorially, with the aid of the transfer matrix method. We give further applications of the 2-rank formulas, including the determination of the nonzeros of certain binary cyclic codes, and a criterion in terms of the trace function to decide for which beta in F-2d* the polynomial x(6) + x + beta has a zero in F-2d, when d is odd. (C) 1999 Academic Press.
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