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

Maximally Recoverable Local Reconstruction Codes (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 h 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.