Rate optimal binary linear locally repairable codes with small availability

Abstract

A locally repairable code with availability has the property that every code symbol can be recovered from multiple, disjoint subsets of other symbols of small size. In particular, a code symbol is said to have (r,t)-availability if it can be recovered from t disjoint subsets, each of size at most r. A code with availability is said to be rate-optimal, if its rate is maximum among the class of codes with given locality, availability, and alphabet size.This paper focuses on rate-optimal binary, linear codes with small availability, and makes four contributions. First, it establishes tight upper bounds on the rate of binary linear codes with (r, 2) and (2, 3) availability. Second, it establishes a uniqueness result for binary rate-optimal codes, showing that for certain classes of binary linear codes with (r, 2) and (2, 3)-availability, any rate optimal code must be a direct sum of shorter rate optimal codes. Third, it presents novel upper bounds on the rates of binary linear codes with (2,t) and (r, 3)-availability. In particular, the main contribution here is a new method for bounding the number of cosets of the dual of a code with availability, using its covering properties. Finally, it presents a class of locally repairable linear codes associated with convex polyhedra, focusing on the codes associated with the Platonic solids. It demonstrates that these codes are locally repairable with t = 2, and that the codes associated with (geometric) dual polyhedra are (coding theoretic) duals of each other.

Erasure coding for distributed storage: an overview

“…The above bound (27), derived in [111], is tighter than (26) as r increases for any fixed t. An upper bound on rate of an (n, k, r = 2, t) SA-LR code over F 2 that for large t, becomes tighter in comparison with the bounds in either (26) or (27), is presented in [144]. Also contained in [144], is an upper bound on the rate of an (n, k, r, 3) SA-LR code over F 2 which is tighter than the bound in either (27) or (26) for r > 72 and which makes use of the "transpose"-based rate equation appearing in [111].…”

“…Since r+t t < (r + 1) t , the code has smaller block length as well. Direct-Sum Construction: It is shown in [144] that the direct sum of m copies of the [7,3] Simplex code yields an SA-LR code with parameters (7m, 3m, 2, 3) having maximum possible rate for n = 7m, r = 2, t = 3, q = 2.…”

Linear programming bounds for distributed storage codes

“…To rewrite the second constraint, notice that N j=1 κ(s j , w j ) = (−1) |s∩w| (q − 1) |w−s| (20) for any pair (s, w) such that |s ∩ w| = i and |w − s| = t − i. In addition, the number of s such that |s ∩ w| = i and |w − s| = t − i is equal to…”

Bounds on the rate and minimum distance of codes with availability

“…(2) on rate of an (n, k, r, 3) sa code using a greedy algorithm. 2) In subsection II-B we derive a bound Eq. (23) and (24) on rate of an (n, k, r, t) sa code for general t using a simple observation on the parity check matrix of a strict availability code and its transpose.…”

“…Therefore there is evidence to suggest that if (n, k, r, t) sa codes are constructable then they are good candidates for best possible availability codes in terms of rate. Discussion of strict availability codes can be found in a recent paper [2].…”