A construction of Maximally Recoverable LRCs for small number of local groups

Maximally Recoverable Local Reconstruction Codes (MR LRCs) are codes designed for distributed storage to provide maximum resilience to failures for a given amount of storage redundancy and locality. An (n, r, h, a, g)-MR LRC has n coordinates divided into g local groups of size r = n/g, where each local group has ‘a’ local parity checks and there are an additional ‘h’ global parity checks. Such a code can correct ‘a’ erasures in each local group and any additional erasures. Constructions of MR LRCs over small fields is desirable since field size determines the encoding and decoding efficiency in practice. In this work, we give a new construction of (n, r, h, a, g)-MR-LRCs over fields of size q = O(n)
^{h+(g−1)a−⌈h/g⌉}
which generalizes a construction of Hu and Yekhanin (ISIT 2016). This improves upon state of the art when there are a small number of local groups, which is true in practical deployments of MR LRCs.