ECCC-Report TR19-005https://eccc.weizmann.ac.il/report/2019/005Comments and Revisions published for TR19-005en-usTue, 28 Sep 2021 17:54:51 +0300
Revision 1
| An Exponential Lower Bound on the Sub-Packetization of MSR Codes |
Omar Alrabiah,
Venkatesan Guruswami
https://eccc.weizmann.ac.il/report/2019/005#revision1An $(n,k,\ell)$-vector MDS code is a $\mathbb{F}$-linear subspace of $(\mathbb{F}^\ell)^n$ (for some field $\mathbb{F}$) of dimension $k\ell$, such that any $k$ (vector) symbols of the codeword suffice to determine the remaining $r=n-k$ (vector) symbols. The length $\ell$ of each codeword symbol is called the sub-packetization of the code. Such a code is called minimum storage regenerating (MSR), if any single symbol of a codeword can be recovered by downloading $\ell/r$ field elements (which is known to be the least possible) from each of the other symbols.
MSR codes are attractive for use in distributed storage systems, and by now a variety of ingenious constructions of MSR codes are available. However, they all suffer from exponentially large sub-packetization of at least $r^{k/r}$. Our main result is an almost tight lower bound showing that for an MSR code, one must have $\ell \ge \exp(\Omega(k/r))$. Previously, a lower bound of $\approx \exp(\sqrt{k/r})$, and a tight lower bound for a restricted class of optimal access MSR codes, were known. Our work settles a key question concerning MSR codes that has received much attention, with a short proof hinging on one key definition that is somewhat inspired by Galois theory.Tue, 28 Sep 2021 17:54:51 +0300https://eccc.weizmann.ac.il/report/2019/005#revision1
Paper TR19-005
| An Exponential Lower Bound on the Sub-Packetization of MSR Codes |
Omar Alrabiah,
Venkatesan Guruswami
https://eccc.weizmann.ac.il/report/2019/005An $(n,k,\ell)$-vector MDS code is a $\mathbb{F}$-linear subspace of $(\mathbb{F}^\ell)^n$ (for some field $\mathbb{F}$) of dimension $k\ell$, such that any $k$ (vector) symbols of the codeword suffice to determine the remaining $r=n-k$ (vector) symbols. The length $\ell$ of each codeword symbol is called the sub-packetization of the code. Such a code is called minimum storage regenerating (MSR), if any single symbol of a codeword can be recovered by downloading $\ell/r$ field elements (which is known to be the least possible) from each of the other symbols.
MSR codes are attractive for use in distributed storage systems, and by now a variety of ingenious constructions of MSR codes are available. However, they all suffer from exponentially large sub-packetization of at least $r^{k/r}$. Our main result is an almost tight lower bound showing that for an MSR code, one must have $\ell \ge \exp(\Omega(k/r))$. Previously, a lower bound of $\approx \exp(\sqrt{k/r})$, and a tight lower bound for a restricted class of optimal access MSR codes, were known. Our work settles a key question concerning MSR codes that has received much attention, with a short proof hinging on one key definition that is somewhat inspired by Galois theory.Wed, 16 Jan 2019 20:36:24 +0200https://eccc.weizmann.ac.il/report/2019/005