An Exponential Lower Bound on the Sub-Packetization of Minimum Storage Regenerating Codes

An <inline-formula> <tex-math notation="LaTeX">$(n,k,\ell)$ </tex-math></inline-formula>-vector MDS code over a field <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}$ </tex-math></inline-formula> is a <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}$ </tex-math></inline-formula>-linear subspace of <inline-formula> <tex-math notation="LaTeX">$(\mathbb {F}^\ell)^{n}$ </tex-math></inline-formula> of dimension <inline-formula> <tex-math notation="LaTeX">$k\ell $ </tex-math></inline-formula>, such that any <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> (vector) symbols of the codeword suffice to determine the remaining <inline-formula> <tex-math notation="LaTeX">$r=n-k$ </tex-math></inline-formula> (vector) symbols. The length <inline-formula> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> of each codeword symbol is called the <italic>sub-packetization</italic> of the code. Such a code is called minimum storage regenerating (MSR), if any single symbol of a codeword can be recovered by downloading <inline-formula> <tex-math notation="LaTeX">$\ell /r$ </tex-math></inline-formula> field elements (which is known to be the minimum 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 <inline-formula> <tex-math notation="LaTeX">$\ell \gtrsim r^{k/r}$ </tex-math></inline-formula>. Our main result is an almost tight lower bound showing that for an MSR code, one must have <inline-formula> <tex-math notation="LaTeX">$\ell \geqslant \exp (\Omega (k/r))$ </tex-math></inline-formula>. Previously, a lower bound of <inline-formula> <tex-math notation="LaTeX">$\approx \exp (\sqrt {k/r})$ </tex-math></inline-formula>, and a tight lower bound for a restricted class of “optimal access” MSR codes, were known.

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