Explicit MBR all-symbol locality codes

Abstract

Node failures are inevitable in distributed storage systems (DSS). To enable efficient repair when faced with such failures, two main techniques are known: Regenerating codes, i.e., codes that minimize the total repair bandwidth; and codes with locality, which minimize the number of nodes participating in the repair process. This paper focuses on regenerating codes with locality, using pre-coding based on Gabidulin codes, and presents constructions that utilize minimum bandwidth regenerating (MBR) local codes. The constructions achieve maximum resilience (i.e., optimal minimum distance) and have maximum capacity (i.e., maximum rate). Finally, the same pre-coding mechanism can be combined with a subclass of fractional-repetition codes to enable maximum resilience and repair-by-transfer simultaneously. I. BACKGROUND A. Vector CodesAn [n, K, d min , α] vector code over a field F q is a code C of block length n, having a symbol alphabet F α q for some α > 1, satisfying the additional property that given c, c ∈ C and a, b ∈ F q , ac + bc also belongs to C. As a vector space over F q , C has dimension K, termed the scalar dimension (equivalently, the file size) of the code and as a code over the alphabet F α q , the code has minimum distance d min . Associated with the vector code C is an F q -linear scalar code C (s) of length N = nα, where C (s) is obtained by expanding each vector symbol within a codeword into α scalar symbols (in some prescribed order). Given a generator matrix G for the scalar code C (s) , the first code symbol in the vector code is naturally associated with the first α columns of G etc. We will refer to the collection of α columns of G associated with the i th code symbol c i as the i th thick column and to avoid confusion, the columns of G themselves as thin columns. B. Locality in Vector CodesLet C be an [n, K, d min , α] vector code over a field F q , possessing a (K × nα) generator matrix G. The i th code symbol, c i , is said to have (r, δ) locality, δ ≥ 2, if there exists a punctured code C i := C| Si of C (called a local code) with support S i ⊆ {1, 2, · · · , n} such thatThe code C is said to have (r, δ) information locality if there exists l code symbols with (r, δ) locality and respective support setsThe code C is said to have (r, δ) all-symbol locality if all code symbols have (r, δ) locality. A code with (r, δ) information (respectively, all-symbol) locality is said to have full (r, δ) information (respectively, all-symbol) locality, if all local codes have parameters given by |S i | = r + δ − 1 and d min (C i ) = δ, for i = 1, · · · , l.The concept of locality for scalar codes, with δ = 2, was introduced in [1] and extended in [2] and [3] to scalar codes with arbitrary δ, and vector codes with δ = 2, respectively. This was further extended to vector codes with arbitrary δ in [4] and [5], where, in addition to constructions of vector codes with locality, authors derive minimum distance upper bounds and also consider settings in which the local codes have regeneration properti...

Codes with locality for two erasures

“…Now, we will show that the sequence {f m } as defined by (49), satisfies the recursion given in (14) and (15), i.e., f r+β = n and…”

“…Then the sets {S i , 1 ≤ i ≤ b = r + β} achieve the upper bounds given in Lemma 5.4 with equality, i.e., ∀m ∈ [b], f m = e m , where f m is as described by (13) and e m is as defined recursively by (14) and (15).…”

Optimal locally repairable codes with local minimum storage regeneration via rank-metric codes

“…A similar construction for vector LRC also allows to design optimal MBR-LRC [30]. On the similar note, following the mechanism given in [19], scalar LRCs presented in this paper can also be used to design novel MBR-LRC.…”

Asymptotically Optimal Regenerating Codes Over Any Field

“…Further aspects of regenerating codes where thoroughly studied in recent years [2,7,8,9,18,21,26]. In particular, the problem of constructing high rate MSR codes, i.e., with a constant number of parity nodes, has received a great deal of attention [4,22,24,29,30].…”