Improved Schemes for Asymptotically Optimal Repair of MDS Codes

We consider $(\text {n}, \text {k}, \text {l})$ MDS codes of length n , dimension k , and subpacketization l over a finite field ${\it\text { F }}$ . A codeword of such a code consists of n column-vectors of length l over ${\it\text { F }}$ , with the property that any k of them suffice to recover the entire codeword. Each of these n vectors may be stored on a separate node in a network. If one of the n nodes fails, we can recover its content by downloading symbols from the surviving nodes, and the total number of symbols downloaded in the worst case is called the repair bandwidth of the code. By the cut-set bound, the repair bandwidth of an $(\text {n},\text {k},\text {l})$ MDS code is at least $\text {l}(\text {n}{-}1)/(\text {n}{-}\text {k})$ . There are several constructions of MDS codes whose repair bandwidth meets or asymptotically meets the cut-set bound. For example, letting $\text {r}=\text {n}-\text {k}$ denote the number of parities, Ye and Barg constructed $(\text {n},\text {k},\text {r}^{\text {n}})$ Reed–Solomon codes that asymptotically meet the cut-set bound, Ye and Barg also constructed optimal-bandwidth and optimal-update $(\text {n},\text {k},\text {r}^{\text {n}})$ MDS codes. Wang, Tamo, and Bruck constructed optimal-bandwidth $(\text {n}, \text {k}, \text {r}^{\text {n}/(\text {r}+1)})$ MDS codes, and these codes have the smallest known subpacketization for optimal-bandwidth MDS codes. A key idea in all these constructions is to represent certain integers in base r . We show how this technique can be refined to improve the subpacketization of the two MDS code constructions by Ye and Barg, while achieving asymptotically optimal repair bandwidth. Specifically, when $\text {r}=\text {s}^{\text {m}}$ for an integer s , we obtain an $(\text {n},\text {k},\text {s}^{\text {m}+\text {n}-1})$ Reed–Solomon code and an optimal-update $(\text {n},\text {k},\text {s}^{\text {m}+\text {n}-1})$ MDS code, both having asymptotically optimal repair bandwidth. Thus for $\text {r} = 2^{\text {m}}$ , for example, we achieve the subpacketization of $2^{\text {m}+\text {n}-1}$ rather than $2^{{\it\text { mn}}}$ in the original constructions by Ye and Barg. When r is not an integral power, we can still obtain $(\text {n},\text {k},\text {s}^{\text {m}+\text {n}-1})$ Reed–Solomon codes and optimal-update $(\text {n},\text {k},\text {s}^{\text {m}+\text {n}-1})$ MDS codes by choosing positive integers s and m such that $\text {s}^{\text {m}} \leqslant r$ . In this case, however, the resulting codes have bandwidth that is near-optimal rather than asymptotically optimal. We also present an extension of this idea to reduce the subpacketization of the Wang–Tamo–Bruck construction while achieving a repair-by-transfer scheme with asymptotically optimal repair bandwidth. For example, for $\text {r} = 2^{\text {m}}$ we achieve the subpacketization of $2^{\text {k}/\text {r}+\text {m}-1}$ , which significantly improves upon the subpacketization of $2^{{\it\text { mn}}/(\text {r}+1)}$ in the Wang–Tamo–Bruck construction. Based on the foregoing examples, we believe our approach may be generally useful in reducing the subpacketization of MDS code constructions that utilize r -ary expansion.