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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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 69; no. 1; pp. 107 - 125
Main Authors: Hou, Hanxu, Han, Yunghsiang S., Lee, Patrick P. C., Wu, You, Han, Guojun, Blaum, Mario
Format: Journal Article
Language:English
Published: New York IEEE 01.01.2023
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
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
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Description
Summary: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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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2022.3202140