The objective of this activity is to construct and study new families of locally recoverable codes.
Locally recoverable codes (LRC) are error correcting codes used in distributed storage systems (e.g. clouds), where any data stored on the databased should be recovered by the users at any time while some servers storing the files may fail or be subject to maintenance operations. A solution to address this issue is to encode the file into a codeword of an LRC. The main feature of these LRC is that, given a codeword of such code, any entry which is erased can be reconstructed as a linear combination of a small amount of other entries.
Our first objective is to focus on Tamo and Bargs construction:
https://arxiv.org/pdf/1311.3284.pdf
and its generalisation using algebraic curves by Barg, Tamo and Vladut:
https://arxiv.org/pdf/1603.08876.pdf
We aim at studying this construction and discussing an abstract setting to design locally recoverable codes. In particular, the previous constructions can be understood in terms of morphism of curves and fibre products. The objective is to discuss various definitions of fibre products for error correcting codes which could permit to address usual issues for LRC’s:
- Locality : each entry of a codeword can be recovered as a linear combination of a small number of other entries;
- availability : for a given entry, how many sets of other entries can be used for recovery;
- what about the parameters : dimension, minimum distance.
For the various possible constructions simulating the notion of fibre products for codes, we aim at testing various examples of constructions using classical code families such as BCH, Reed-Muller or algebraic geometry codes. We expect to run experiments using a computer algebra software such as Sage.
