Fractional Repetition Codes With Flexible Repair From Combinatorial Designs

Abstract

Fractional repetition (FR) codes are a class of regenerating codes for distributed storage systems with an exact (table-based) repair process that is also uncoded, i.e., upon failure, a node is regenerated by simply downloading packets from the surviving nodes. In our work, we present constructions of FR codes based on Steiner systems and resolvable combinatorial designs such as affine geometries, Hadamard designs and mutually orthogonal Latin squares. The failure resilience of our codes can be varied in a simple manner. We construct codes with normalized repair bandwidth (β) strictly larger than one; these cannot be obtained trivially from codes with β = 1. Furthermore, we present the Kronecker product technique for generating new codes from existing ones and elaborate on their properties. FR codes with locality are those where the repair degree is smaller than the number of nodes contacted for reconstructing the stored file. For these codes we establish a tradeoff between the local repair property and failure resilience and construct codes that meet this tradeoff. Much of prior work only provided lower bounds on the FR code rate. In our work, for most of our constructions we determine the code rate for certain parameter ranges.

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

“…We assume β = 1 for the code construction throughout paper as in [11], which can be simply expanded to the case of β > 1. It is noted that this expansion does not cover all the FR codes for β > 1 [18]. For MBR codes with β = 1, d is equal to α, and the number of data symbols to be stored in DSS is given as:…”

“…If there exists at least one resolution, then the code is called a resolvable FR code. It is clear that the FR codes from the first construction are not resolvable because q is not a factor of q 2 − 1, while the FR codes from the net in [18] are always resolvable. This is evidence that the proposed construction is not a proper subset of the constructions in [18].…”

“…For ζ = 1, we have an RDS with parameters (8, 6, 7, 1) as D = {0, 18,22,28,29,31, 43}, and its characteristic sequence is given as:…”

“…In [16], the existence of FR codes for each parameter set is shown by algorithm-based search, and some examples for each parameter set are provided. Many kinds of FR codes have been constructed mostly based on algebraic structures, combinatorial designs and graph theory [15,[17][18][19][20][21][22]. In [15], Steiner systems are used as a class of (v, (v − 1)/(k − 1), k) FR codes with β = 1 under the existence of Steiner system S(2, k, v).…”

“…These FR codes are a subclass of the FR codes from Steiner systems in [15] and additionally designed to have a scalable property. In [18], various constructions of resolvable FR codes, especially net FR codes, are proposed by using grids, affine resolvable designs, Hadamard designs and mutually orthogonal Latin squares. The corresponding constructable parameters are (2a, a, 2) with β = 1 from grids, (qρ, q m−1 , 1 ≤ ρ ≤ (q m − 1)/(q − 1)) with β = q m−2 from affine resolvable designs, (8a − 2, 2a, 4a − 1) with β = a from Hadamard designs (where 4a − 1 ≥ 7 is an odd prime power) and (ρp m , p m , ρ ≤ p m − 1) or (4a, a, 4) with β = 1 from mutually orthogonal Latin squares (for a prime p, positive integers m and an integer a = 2, 6).…”