A Generalization of Array Codes With Local Properties and Efficient Encoding/Decoding
An <inline-formula> <tex-math notation="LaTeX">(n,k) </tex-math></inline-formula> recoverable property array code is composed of <inline-formula> <tex-math notation="LaTeX">m\times n </tex-math></inline-formula> arrays such that any...
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| Vydané v: | IEEE transactions on information theory Ročník 69; číslo 1; s. 107 - 125 |
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| Hlavní autori: | , , , , , |
| Médium: | Journal Article |
| Jazyk: | English |
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New York
IEEE
01.01.2023
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
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| Abstract | An <inline-formula> <tex-math notation="LaTeX">(n,k) </tex-math></inline-formula> recoverable property array code is composed of <inline-formula> <tex-math notation="LaTeX">m\times n </tex-math></inline-formula> arrays such that any <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> out of <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> columns suffice to retrieve all the information symbols, where <inline-formula> <tex-math notation="LaTeX">n > k </tex-math></inline-formula>. Note that maximum distance separable (MDS) array code is a special <inline-formula> <tex-math notation="LaTeX">(n,k) </tex-math></inline-formula> recoverable property array code of size <inline-formula> <tex-math notation="LaTeX">m\times n </tex-math></inline-formula> with the number of information symbols being <inline-formula> <tex-math notation="LaTeX">km </tex-math></inline-formula>. Expanded-Blaum-Roth (EBR) codes and Expanded-Independent-Parity (EIP) codes are two classes of <inline-formula> <tex-math notation="LaTeX">(n,k) </tex-math></inline-formula> recoverable property array codes that can repair any one symbol in a column by locally accessing some other symbols within the column, where the number of symbols <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> in a column is a prime number. By generalizing the constructions of EBR and EIP codes, we propose new <inline-formula> <tex-math notation="LaTeX">(n,k) </tex-math></inline-formula> recoverable property array codes, such that any one symbol can be locally recovered and the number of symbols in a column can be not only a prime number but also a power of an odd prime number. Also, we present an efficient encoding/decoding method for the proposed generalized EBR (GEBR) and generalized EIP (GEIP) codes based on the LU factorization of a Vandermonde matrix. We show that the proposed decoding method has less computational complexity than existing methods. Furthermore, we show that the proposed GEBR codes have both a larger minimum symbol distance and a larger recovery ability of erased lines for some parameters when compared to EBR codes. We also present a necessary and sufficient condition of enabling EBR codes to recover any <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula> erased lines of a slope for any parameter <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula>, which was an open problem. Moreover, we show that EBR codes can recover any <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula> consecutive erased lines of any slope for any parameter <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula>. |
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| AbstractList | An [Formula Omitted] recoverable property array code is composed of [Formula Omitted] arrays such that any [Formula Omitted] out of [Formula Omitted] columns suffice to retrieve all the information symbols, where [Formula Omitted]. Note that maximum distance separable (MDS) array code is a special [Formula Omitted] recoverable property array code of size [Formula Omitted] with the number of information symbols being [Formula Omitted]. Expanded-Blaum-Roth (EBR) codes and Expanded-Independent-Parity (EIP) codes are two classes of [Formula Omitted] recoverable property array codes that can repair any one symbol in a column by locally accessing some other symbols within the column, where the number of symbols [Formula Omitted] in a column is a prime number. By generalizing the constructions of EBR and EIP codes, we propose new [Formula Omitted] recoverable property array codes, such that any one symbol can be locally recovered and the number of symbols in a column can be not only a prime number but also a power of an odd prime number. Also, we present an efficient encoding/decoding method for the proposed generalized EBR (GEBR) and generalized EIP (GEIP) codes based on the LU factorization of a Vandermonde matrix. We show that the proposed decoding method has less computational complexity than existing methods. Furthermore, we show that the proposed GEBR codes have both a larger minimum symbol distance and a larger recovery ability of erased lines for some parameters when compared to EBR codes. We also present a necessary and sufficient condition of enabling EBR codes to recover any [Formula Omitted] erased lines of a slope for any parameter [Formula Omitted], which was an open problem. Moreover, we show that EBR codes can recover any [Formula Omitted] consecutive erased lines of any slope for any parameter [Formula Omitted]. An <inline-formula> <tex-math notation="LaTeX">(n,k) </tex-math></inline-formula> recoverable property array code is composed of <inline-formula> <tex-math notation="LaTeX">m\times n </tex-math></inline-formula> arrays such that any <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> out of <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> columns suffice to retrieve all the information symbols, where <inline-formula> <tex-math notation="LaTeX">n > k </tex-math></inline-formula>. Note that maximum distance separable (MDS) array code is a special <inline-formula> <tex-math notation="LaTeX">(n,k) </tex-math></inline-formula> recoverable property array code of size <inline-formula> <tex-math notation="LaTeX">m\times n </tex-math></inline-formula> with the number of information symbols being <inline-formula> <tex-math notation="LaTeX">km </tex-math></inline-formula>. Expanded-Blaum-Roth (EBR) codes and Expanded-Independent-Parity (EIP) codes are two classes of <inline-formula> <tex-math notation="LaTeX">(n,k) </tex-math></inline-formula> recoverable property array codes that can repair any one symbol in a column by locally accessing some other symbols within the column, where the number of symbols <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> in a column is a prime number. By generalizing the constructions of EBR and EIP codes, we propose new <inline-formula> <tex-math notation="LaTeX">(n,k) </tex-math></inline-formula> recoverable property array codes, such that any one symbol can be locally recovered and the number of symbols in a column can be not only a prime number but also a power of an odd prime number. Also, we present an efficient encoding/decoding method for the proposed generalized EBR (GEBR) and generalized EIP (GEIP) codes based on the LU factorization of a Vandermonde matrix. We show that the proposed decoding method has less computational complexity than existing methods. Furthermore, we show that the proposed GEBR codes have both a larger minimum symbol distance and a larger recovery ability of erased lines for some parameters when compared to EBR codes. We also present a necessary and sufficient condition of enabling EBR codes to recover any <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula> erased lines of a slope for any parameter <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula>, which was an open problem. Moreover, we show that EBR codes can recover any <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula> consecutive erased lines of any slope for any parameter <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula>. |
| Author | Lee, Patrick P. C. Blaum, Mario Wu, You Han, Guojun Han, Yunghsiang S. Hou, Hanxu |
| Author_xml | – sequence: 1 givenname: Hanxu orcidid: 0000-0001-7328-9341 surname: Hou fullname: Hou, Hanxu email: houhanxu@163.com organization: School of Electrical Engineering and Intelligentization, Dongguan University of Technology, Dongguan, China – sequence: 2 givenname: Yunghsiang S. orcidid: 0000-0002-3592-1681 surname: Han fullname: Han, Yunghsiang S. email: yunghsiang@gmail.com organization: Shenzhen Institute for Advanced Study, University of Electronic Science and Technology of China, Shenzhen, China – sequence: 3 givenname: Patrick P. C. orcidid: 0000-0002-4501-4364 surname: Lee fullname: Lee, Patrick P. C. email: pclee@cse.edu.cuhk.hk organization: Department of Computer Science and Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong – sequence: 4 givenname: You surname: Wu fullname: Wu, You email: 278008313@qq.com organization: Beijing Didi Infinity Technology and Development Company Ltd., Beijing, China – sequence: 5 givenname: Guojun surname: Han fullname: Han, Guojun email: gjhan@gdut.edu.cn organization: School of Information Engineering, Guangdong University of Technology, Guangzhou, China – sequence: 6 givenname: Mario orcidid: 0000-0002-5711-9411 surname: Blaum fullname: Blaum, Mario email: mblaum@hotmail.com organization: IBM Research Division-Almaden, San Jose, CA, USA |
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| SubjectTerms | Array codes Arrays Codes Data centers Decoding efficient encoding/decoding Encoding Encoding-Decoding expanded-Blaum-Roth codes expanded-independent-parity codes Information retrieval local repair Maintenance engineering Parameters Prime numbers Symbols |
| Title | A Generalization of Array Codes With Local Properties and Efficient Encoding/Decoding |
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