Abstract
Fractional repetition (FR) codes are a class of distributed storage codes that replicate and distribute information data over several nodes for easy repair, as well as efficient reconstruction. In this paper, we propose three new constructions of FR codes based on relative difference sets (RDSs) with λ = 1 . Specifically, we propose new ( q 2 - 1 , q , q ) FR codes using cyclic RDS with parameters ( q + 1 , q - 1 , q , 1 ) constructed from q-ary m-sequences of period q 2 - 1 for a prime power q, ( p 2 , p , p ) FR codes using non-cyclic RDS with parameters ( p , p , p , 1 ) for an odd prime p or p = 4 and ( 4 l , 2 l , 2 l ) FR codes using non-cyclic RDS with parameters ( 2 l , 2 l , 2 l , 1 ) constructed from the Galois ring for a positive integer l. They are differentiated from the existing FR codes with respect to the constructable code parameters. It turns out that the proposed FR codes are (near) optimal for some parameters in terms of the FR capacity bound. Especially, ( 8 , 3 , 3 ) and ( 9 , 3 , 3 ) FR codes are optimal, that is, they meet the FR capacity bound for all k. To support various code parameters, we modify the proposed ( q 2 - 1 , q , q ) FR codes using decimation by a factor of the code length q 2 - 1 , which also gives us new good FR codes.
Highlights
As users of social media services and cloud services frequently upload large data files such as images and videos, huge storage space is required, which is implemented in the form of distributed storage systems (DSSs) [1,2]
It is necessary to find a new class of node failure-handling protocols that is well-fitted for the DSS environment, and for this reason, locally repairable codes [3,4,5,6,7]
We show via theoretical derivations and numerical analysis that the proposed fractional repetition (FR) codes are optimal for some parameters in terms of the FR capacity bound
Summary
As users of social media services and cloud services frequently upload large data files such as images and videos, huge storage space is required, which is implemented in the form of distributed storage systems (DSSs) [1,2]. The FR codes constructed from Steiner systems [15] support the number of data symbols equal to or slightly larger than the MBR capacity in (1). This means that the file size of the FR codes from. We first propose new three constructions of (q2 − 1, q, q), ( p2 , p, p) and (4l , 2l , 2l ) FR codes based on relative difference sets with parameters (q + 1, q − 1, q, 1), ( p, p, p, 1) and (2l , 2l , 2l , 1), respectively, where q is a prime power, p is an odd prime or p = 4 and l is a positive integer.
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