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

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 (q2 − 1, q, q) FR codes using cyclic RDS with parameters (q + 1, q− 1, q, 1) constructed from q-ary m-sequences of period q2 − 1 for a prime power q, (p2, p, p) FR codes using non-cyclic RDS with parameters (p, p, p, 1) for an odd prime p or p = 4 and (4l , 2l , 2l) FR codes using non-cyclic RDS with parameters (2l , 2l , 2l , 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 (q2 − 1, q, q) FR codes using decimation by a factor of the code length q2 − 1, which also gives us new good FR codes.