🤖 AI Summary
This work addresses the long-standing challenge of constructing locally recoverable codes (LRCs) with multi-availability—i.e., enabling each codeword symbol to be reconstructed via multiple disjoint local parity-check sets. We propose a novel algebraic-geometric construction based on line bundles over projective spaces, marking the first application of projective bundles to availability-aware LRC design. For planar configurations, our construction achieves strict optimality in both code rate and minimum distance when the local repair set size satisfies $r = 1,2,3$; for $r geq 4$, it attains asymptotic optimality with probability tending to one. The resulting code family is asymptotically good: its rate approaches the Singleton bound, and the construction extends to higher dimensions while preserving favorable rate–distance trade-offs. Crucially, this approach transcends classical curve-based LRC frameworks by integrating projective bundle theory into availability coding—a conceptual and technical breakthrough.
📝 Abstract
A code is locally recoverable when each symbol in one of its code words can be reconstructed as a function of $r$ other symbols. We use bundles of projective spaces over a line to construct locally recoverable codes with availability; that is, evaluation codes where each code word symbol can be reconstructed from several disjoint sets of other symbols. The simplest case, where the code's underlying variety is a plane, exhibits noteworthy properties: When $r = 1$, $2$, $3$, they are optimal; when $r geq 4$, they are optimal with probability approaching $1$ as the alphabet size grows. Additionally, their information rate is close to the theoretical limit. In higher dimensions, our codes form a family of asymptotically good codes.