A Rate-Optimal Construction of Codes with Sequential Recovery with Low Block Length

An erasure code is said to be a code with sequential recovery with parameters r and t, if for any s ≤ t erased code symbols, there is an s-step recovery process in which at each step we recover exactly one erased code symbol by contacting at most r other code symbols. In earlier work by the same authors, presented at ISIT 2017, we had given a construction for binary codes with sequential recovery from t erasures, with locality parameter r, which were optimal in terms of code rate for given r, t, but where the block length was large, on the order of r t , for some constant c > 1. In the present paper, we present an alternative construction of a rate-optimal code for any value of t and any r ≥ 3, where the block length is significantly smaller, on the order of r 5t 4 + 7 4 (in some instances of order r 3t 2 ). Our construction is based on the construction of certain kind of treelike graphs with girth t+1. We construct these graphs and hence the codes recursively.