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Construction of New Fractional Repetition Codes from Relative Difference Sets with λ=1

by Young-Sik Kim 1,†, Hosung Park 2,*,† and Jong-Seon No 3,†
1
Department of Information and Communication Engineering, Chosun University, Gwangju 61452, Korea
2
School of Electronics and Computer Engineering, Chonnam National University, Gwangju 61186, Korea
3
Department of Electrical and Computer Engineering, Institute of New Media and Communications, Seoul National University, Seoul 08826, Korea
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Entropy 2017, 19(10), 563; https://doi.org/10.3390/e19100563
Received: 12 August 2017 / Revised: 19 October 2017 / Accepted: 19 October 2017 / Published: 22 October 2017
(This article belongs to the Section Information Theory, Probability and Statistics)
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. View Full-Text
Keywords: distributed storage systems (DSS); fractional repetition (FR) codes; FR capacity; minimum bandwidth regenerating (MBR) codes; relative difference sets (RDSs); q-ary m-sequences distributed storage systems (DSS); fractional repetition (FR) codes; FR capacity; minimum bandwidth regenerating (MBR) codes; relative difference sets (RDSs); q-ary m-sequences
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Kim, Y.-S.; Park, H.; No, J.-S. Construction of New Fractional Repetition Codes from Relative Difference Sets with λ=1. Entropy 2017, 19, 563.

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