Locally recoverable code


Locally recoverable codes are a family of error correction codes that were introduced first by D. S. Papailiopoulos and A. G. Dimakis and have been widely studied in information theory due to their applications related to distributive and cloud storage systems.
An LRC is an linear code such that there is a function that takes as input and a set of other coordinates of a codeword different from, and outputs.

Overview

Erasure-correcting codes, or simply erasure codes, for distributed and cloud storage systems, are becoming more and more popular as a result of the present spike in demand for cloud computing and storage services. This has inspired researchers in the fields of information and coding theory to investigate new facets of codes that are specifically suited for use with storage systems.

It is well-known that LRC is a code that needs only a limited set of other symbols to be accessed in order to restore every symbol in a codeword. This idea is very important for distributed and cloud storage systems since the most common error case is when one storage node fails. The main objective is to recover as much data as possible from the fewest additional storage nodes in order to restore the node. Hence, Locally Recoverable Codes are crucial for such systems.
The following definition of the LRC follows from the description above: an -Locally Recoverable Code of length is a code that produces an -symbol codeword from information symbols, and for any symbol of the codeword, there exist at most other symbols such that the value of the symbol can be recovered from them. The locality parameter satisfies because the entire codeword can be found by accessing symbols other than the erased symbol. Furthermore, Locally Recoverable Codes, having the minimum distance, can recover erasures.

Definition

Let be a linear code. For, let us denote by the minimum number of other coordinates we have to look at to recover an erasure in coordinate. The number is said to be the locality of the -th coordinate of the code. The locality of the code is defined as

An locally recoverable code is an linear code with locality.
Let be an -locally recoverable code. Then an erased component can be recovered linearly, i.e. for every, the space of linear equations of the code contains elements of the form, where.

Optimal locally recoverable codes

Theorem Let and let be an -locally recoverable code having disjoint locality sets of size. Then

An -LRC is said to be optimal if the minimum distance of satisfies

Tamo–Barg codes

Let be a polynomial and let be a positive integer. Then is said to be -good if
We say that is a splitting covering for.

Tamo–Barg construction

The Tamo–Barg construction utilizes good polynomials.

Example of Tamo–Barg construction

We will use to construct -LRC. Notice that the degree of this polynomial is 5, and it is constant on for, where,,,,,,, and :,,,,,,,. Hence, is a -good polynomial over by the definition. Now, we will use this polynomial to construct a code of dimension and length over. The locality of this code is 4, which will allow us to recover a single server failure by looking at the information contained in at most 4 other servers.
Next, let us define the encoding polynomial:, where. So, .
Thus, we can use the obtained encoding polynomial if we take our data to encode as the row vector . Encoding the vector to a length 15 message vector by multiplying by the generator matrix
For example, the encoding of information vector gives the codeword.
Observe that we constructed an optimal LRC; therefore, using the Singleton bound, we have that the distance of this code is. Thus, we can recover any 6 erasures from our codeword by looking at no more than 8 other components.

Locally recoverable codes with availability

A code has all-symbol locality and availability if every code symbol can be recovered from disjoint repair sets of other symbols, each set of size at most symbols. Such codes are called -LRC.
Theorem The minimum distance of -LRC having locality and availability satisfies the upper bound

If the code is systematic and locality and availability apply only to its information symbols, then the code has information locality and availability, and is called -LRC.
Theorem The minimum distance of an linear -LRC satisfies the upper bound