A Rate-Optimal Construction of Codes with Sequential Recovery with Low Block Length
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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^c^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+2). Our construction is based on the construction of certain kind of tree-like graphs with girth t+1. We construct these graphs and hence the codes recursively.
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