Constructions of Optimal Cyclic $({r},{\delta })$ Locally Repairable Codes

Abstract

A code is said to be a r-local locally repairable code (LRC) if each of its coordinates can be repaired by accessing at most r other coordinates. When some of the r coordinates are also erased, the r-local LRC can not accomplish the local repair, which leads to the concept of (r, δ)-locality. A q-ary [n, k] linear code C is said to have (r, δ)-locality (δ ≥ 2) if for each coordinate i, there exists a punctured subcode of C with support containing i, whose length is at most r + δ − 1, and whose minimum distance is at least δ. The (r, δ)-LRC can tolerate δ − 1 erasures in total, which degenerates to a r-local LRC when δ = 2. A q-ary (r, δ) LRC is called optimal if it meets the Singleton-like bound for (r, δ)-LRCs. A class of optimal q-ary cyclic r-local LRCs with lengths n | q − 1 were constructed by Tamo, Barg, Goparaju and Calderbank based on the q-ary Reed-Solomon codes. In this paper, we construct a class of optimal q-ary cyclic (r, δ)-LRCs (δ ≥ 2) with length n | q − 1, which generalizes the results of Tamo et al. Moreover, we construct a new class of optimal q-ary cyclic r-local LRCs with lengths n | q + 1 and a new class of optimal q-ary cyclic (r, δ)-LRCs (δ ≥ 2) with lengths n | q + 1. The constructed optimal LRCs with length n = q + 1 have the best-known length q + 1 for the given finite field with size q when the minimum distance is larger than 4.

A Tight Rate Bound and Matching Construction for Locally Recoverable Codes With Sequential Recovery From Any Number of Multiple Erasures

“…columns of the generator matrix of the code C| Si has rank equal to dim(C| Si ) (≤ r). Upper bounds on minimum distance and optimal constructions having field size of O(n) for codes with (r, δ) locality can be found in [9], [26], [10], [27]. Codes with (r, δ) locality can guarantee local recovery from any t erasures, if one chooses δ ≥ t + 1.…”

Locally recoverable J-affine variety codes

“…Parameters are showed in Table 8. The defining set of our first code is ∆ = I (0,0) ∪ I (0,1) ∪ I (1,1) ∪ I (2,0) ∪ I (3,0) , and the remaining ones are obtained by successively adding to ∆ the following cyclotomic sets: I (4,0) , I (5,0) , and both I (4,1) and I (6,1) .…”

“…The defining set of our first code is ∆ = I (0,0) ∪ I (0,1) ∪ I (1,0) ∪ I (2,0) . We get the remaining ones by successively adding to ∆ the following cyclotomic sets: I (3,0) , I (4,0) , I (5,0) , both I (1,1) and I (6,0) , and I (7,1) . Example 5.…”

“…Set, in this new example, p = 2, ℓ = 12, s = 3, N 1 = 8 and N 2 = 6. Our codes are defined by the sets I (0,1) , I (0,1) ∪ I (1,1) and I (0,1) ∪ I (1,1) ∪ I (2,1) . Parameters are displayed in Table 10.…”

“…Among the different classes of codes used as good candidates for local recovering, cyclic codes and subfield-subcodes of cyclic codes play an important role, as the cyclic shifts of a recovery set provide again recovery sets [1,10,12,17]. In this article we continue this line of research by using the very general language of affine variety codes.…”