Locally recoverable codes from elliptic surfaces with availability and hierarchical locality
摘要
In this paper, we propose several constructions of Locally Recoverable Codes from elliptic surfaces. In particular, we are able to obtain codes with availability $t>2$, codes with hierarchical locality and, finally, codes which combine availability and hierarchical locality. Our constructions rely on the properties of the torsion groups of elliptic curves and on the fibered structure of elliptic surfaces. In particular, the geometry of the surface is used to introduce a multi-dimensional setting, allowing for more recovery sets, eventually nested one within another.
相关性判断
highThe paper is explicitly about locally recoverable codes, availability, and hierarchical locality, with constructions from elliptic surfaces and elliptic curves, which are directly within coding theory and adjacent algebraic geometry for information theory review.
Directly relevant to cs.IT coding theory with high-confidence relevance analysis and explicit focus on LRCs, availability t>2, and hierarchical locality. Structure analysis indicates substantial technical machinery from elliptic surfaces, torsion groups, isogenies, Riemann-Roch spaces, and AG-code evaluation. Potential value is strongest for algebraic constructions of storage codes, especially combinations of availability and hierarchical locality.
核心问题与主要方法
核心问题
Build locally recoverable codes on elliptic surfaces with stronger local repair features: high availability, hierarchical locality, and both together.
场景：Algebraic-geometry code constructions over finite fields using elliptic curves, elliptic surfaces, torsion points, generic fibers, and specialization maps.
主要方法
For high availability, each smooth elliptic fiber contributes an m-torsion group isomorphic to Z/m x Z/m; lines through a point in this grid give multiple recovery sets whose intersection is only the erased coordinate. Abel's Theorem is used to justify recovery from line-supported subsets on elliptic fibers by controlling divisors and evaluations of functions in H^0(E_gamma, O((m-1)infinity_gamma)). For hierarchical locality, the construction uses quotient isogenies E -> E1 -> E2, rational functions f and g separating points or local fibers, and interpolation on nested fibers/cosets. For elliptic surfaces, curve-level constructions are transferred through the generic fiber E/F_q(B), then specialized to good fibers while excluding bad reduction, poles, and zeros of separating functions. The combined construction uses a torsion factorization m=v*w and nested cosets from E[v] and E[w] to create middle and lower codes with availability 2 at hierarchy levels.
关键贡献与后续阅读
关键贡献
Introduces LRC constructions from elliptic surfaces achieving availability t=m>2 via m-torsion grids on elliptic fibers. Builds hierarchical LRCs on elliptic surfaces by first constructing quotient-isogeny hierarchies on elliptic curves and then lifting them through the generic fiber and specialization maps. Combines availability and hierarchical locality on elliptic surfaces, with the payload claiming this coexistence on higher-dimensional varieties was previously unexplored in this setting. Designs evaluation spaces using Riemann-Roch spaces and towers of quotients to control local interpolation and global parameter bounds. Provides explicit parameter bounds for constructed codes, including length, dimension, and minimum-distance lower bounds for the surface-based availability and hierarchy constructions.
研究启发
How do the resulting parameter ranges compare numerically with existing LRC/HLRC constructions from curves or fibered products for realistic finite fields? Can the bad-fiber and separating-function exclusions be made explicit for concrete elliptic surfaces, rather than handled by existence and asymptotic field-size arguments? Are there worked examples or computational instances showing achievable n, k, d, locality, and availability values?
限制与不确定性
Practical impact is uncertain because parameter quality appears dependent on field size, divisors, bad fibers, and possible field extensions. Abstract and structure evidence suggest constructions and bounds, but not necessarily optimized or competitive parameters.
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