On the Duality and File Size Hierarchy of Fractional Repetition Codes

Abstract

Distributed storage systems that deploy erasure codes can provide better features such as lower storage overhead and higher data reliability. In this paper, we focus on fractional repetition (FR) codes, which are a class of storage codes characterized by the features of uncoded exact repair and minimum repair bandwidth. We study the duality of FR codes, and investigate the relationship between the supported file size of an FR code and its dual code. Based on the established relationship, we derive an improved dual bound on the supported file size of FR codes. We further show that FR codes constructed from t-designs are optimal when the size of the stored file is sufficiently large. Moreover, we present the tensor product technique for combining FR codes, and elaborate on the file size hierarchy of resulting codes.

On the Optimal Minimum Distance of Fractional Repetition Codes

“…It is shown in [16] that the upper bound in (41) can be tighter than the original bound in (3) for certain code parameters. Thus, a safe conclusion can be obtained by taking the minimum of the two bounds.…”

“…Since the roles of points and blocks are reversed in I t , it follows that C t forms a (θ, ρ, α)-FR code [16].…”

“…In [16], the complementary supported file size of an (n, α, ρ)-FR code C, denoted by N k (C), is defined as follows…”

“…Lemma 1. ( [16]) Let C be an (n, α, ρ)-FR code with θ = nα ρ , and let C t be the dual code. Then, the supported file size of C can be determined as…”

“…In [13], Olmez and Ramamoorthy investigated constructions of FR codes based on resolvable combinatorial designs including affine geometries, Hadamard designs, and mutually orthogonal Latin squares. Constructions of new FR codes from existing codes can be obtained by using techniques such as Kronecker product [13], tensor product [16], and symbol extension [20].…”