Dimension Expanders via Rank Condensers

An emerging theory of “linear-algebraic pseudorandomness” aims to understand the linear-algebraic analogs of fundamental Boolean pseudorandom objects where the rank of subspaces plays the role of the size of subsets. In this work, we study and highlight the interrelationships between several such algebraic objects such as subspace designs, dimension expanders, seeded rank condensers, two-source rank condensers, and rank-metric codes. In particular, with the recent construction of near-optimal subspace designs by Guruswami and Kopparty [GK13] as a starting point, we construct good (seeded) rank condensers (both lossless and lossy versions), which are a small collection of linear maps Fn → Ft for t n such that for every subset of Fn of small rank, its rank is preserved (up to a constant factor in the lossy case) by at least one of the maps. We then compose a tensoring operation with our lossy rank condenser to construct constant-degree dimension expanders over polynomially large �...

• Type: Journal
• Year: 2015
• Where: APPROX
• Authors: Michael A. Forbes, Venkatesan Guruswami
• Keywords: Algorithms, APPROX 2015