On the Existence of Optimal Exact-Repair MDS Codes for Distributed Storage

The high repair cost of (n,k) Maximum Distance Separable (MDS) erasure codes has recently motivated a new class of codes, called Regenerating Codes, that optimally trade off storage cost for repair bandwidth. In this paper, we address bandwidth-optimal (n,k,d) Exact-Repair MDS codes, which allow for any failed node to be repaired exactly with access to arbitrary d survivor nodes, where k<=d<=n-1. We show the existence of Exact-Repair MDS codes that achieve minimum repair bandwidth (matching the cutset lower bound) for arbitrary admissible (n,k,d), i.e., k<n and k<=d<=n-1. Our approach is based on interference alignment techniques and uses vector linear codes which allow to split symbols into arbitrarily small subsymbols.Comment: 20 pages, 6 figure

Supersymmetric AdS6AdS_6 black holes from matter coupled F(4)F(4) gauged supergravity

In matter coupled $F(4)$ gauged supergravity in six dimensions, we study supersymmetric $AdS_6$ black holes with various horizon geometries. We find new $AdS_2\,\times\,\Sigma_{\mathfrak{g}_1}\times\Sigma_{\mathfrak{g}_2}$ horizons with $\mathfrak{g}_1>1$ and $\mathfrak{g}_2>1$, and present the black hole solution numerically. The full black hole is an interpolating geometry between the asymptotically $AdS_6$ boundary and the $AdS_2\,\times\,\Sigma_{\mathfrak{g}_1}\times\Sigma_{\mathfrak{g}_2}$ horizon. We also find black holes with horizons of K\"ahler four-cycles in Calabi-Yau fourfolds and Cayley four-cycles in $Spin(7)$ manifolds.Comment: v3: 24 pages, 3 figures. Title changed. Previous title was "D4-branes wrapped on supersymmetric four-cycles from matter coupled $F(4)$ gauged supergravity". Numerical black hole solutions added. Published version. v4: Omitted citations of [29] in the introduction and above (3.21) recovered. arXiv admin note: text overlap with arXiv:1809.0351