Outer bounds on the storage-repair bandwidth trade-off of exact-repair regenerating codes

In this paper, three outer bounds on the normalised storage-repair bandwidth trade-off of regenerating codes having parameter set {(n, k, d),(alpha, beta)} under the exact-repair (ER) setting are presented. The first outer bound, termed as the repair-matrix bound, is applicable for every parameter set (n, k, d), and in conjunction with a code construction known as improved layered codes, it characterises the normalised ER trade-off for the case (n, k = 3, d = n - 1). The bound shows that a non-vanishing gap exists between the ER and functional-repair (FR) trade-offs for every (n, k, d). The second bound, termed as the improved Mohajer-Tandon bound, is an improvement upon an existing bound due to Mohajer et al. and performs better in a region away from the minimum-storage-regenerating (MSR) point. However, in the vicinity of the MSR point, the repair-matrix bound outperforms the improved Mohajer-Tandon bound. The third bound is applicable to linear codes for the case k = d. In conjunction with the class of layered codes, the third outer bound characterises the normalised ER trade-off in the case of linear codes when k = d = n - 1

Explicit MSR Codes with Optimal Access, Optimal Sub-Packetization and Small Field Size for d = k+1, k+2, k+3

This paper presents the construction of an explicit, optimal-access, high-rate MSR code for any (n, k, d = k + 1, k + 2, k + 3) parameters over the finite field F-Q having sub-packetization alpha = q(inverted right perpendicular)n/q(inverted left perpendicular), where q = d - k + 1 and Q = O(n). The sub-packetization of the current construction meets the lower bound proven in a recent work by Balaji et al. in 1]. To our understanding the codes presented in this paper are the first explicit constructions of MSR codes with d < (n - 1) having optimal sub-packetization, optimal access and small field size

Binary, Shortened Projective Reed Muller Codes for Coded Private Information Retrieval

The notion of a Private Information Retrieval (PIR) code was recently introduced by Fazeli, Vardy and Yaakobi 1] who showed that this class of codes permit PIR at reduced levels of storage overhead in comparison with rephcated-server PIR. In the present paper, the construction of an (n, k) tau-server binary linear PIR code having parameters n = Sigma (l)(i=0) ((m)(i)), k = ((m)(l)) and tau = 2(l) for any integer m >= l >= 0 is presented. These codes are obtained through homogeneous-polynomial evaluation and correspond to the binary. Projective Reed Muller (PRM) code. The construction can be extended to yield PIR codes for any tau is an element of{2(l), 2(l) - 1 vertical bar l is an element of Z, l >= 0} and any value of k, through a combination of single-symbol puncturing and shortening of the PRM code. Each of these code constructions above, have smaller storage overhead in comparison with known short block length codes in 1]. For the particular case of tau = 3,4, we show that the codes constructed here are optimal, systematic PIR codes by providing an improved lower bound on the block length n(k, tau) of a systematic PIR code. It follows from a result by Vardy and Yaakobi 2], that these codes also yield optimal, systematic primitive multi-set (n, k, tau)(B) batch codes for tau = 3,4. The PIR code constructions presented here also yield upper bounds on the generahzed Hamming weights of binary PRM codes