The subspace design property for additive codes is a higher-dimensional generalization of the minimum distance property. As shown recently by Brakensiek, Chen, Dhar and Zhang, it implies that the code has similar performance as random linear codes with respect to all "local properties". Explicit algebraic codes, such as folded Reed-Solomon and multiplicity codes, are known to have the subspace design property, but they need alphabet sizes that grow as a large polynomial in the block length. Constructing explicit constant-alphabet subspace design codes was subsequently posed as an open question in Brakensiek, Chen, Dhar and Zhang. In this work, we answer their question and give explicit constructions of subspace design codes over constant-sized alphabets, using the expander-based Alon-Edmonds-Luby (AEL) framework. This generalizes the recent work of Jeronimo and Shagrithaya, which showed that such codes share local properties of random linear codes. Our work obtains this consequence in a unified manner via the subspace design property. In addition, our approach yields some improvements in parameters for list-recovery.

翻译：加法码的子空间设计性质是最小距离性质的高维推广。正如Brakensiek、Chen、Dhar和Zhang最新研究所表明，该性质能够使得码字在"所有局部性质"方面具有与随机线性码相当的性能。已知折叠里德-所罗门码和多重码等显式代数编码具有子空间设计性质，但其字母表大小需要随码长呈高次多项式增长。Brakensiek、Chen、Dhar和Zhang随后将"构造显式常量字母表子空间设计码"列为开放问题。本文解决了该问题，基于扩展器驱动的Alon-Edmonds-Luby（AEL）框架，给出了常量大小字母表上子空间设计码的显式构造。这推广了Jeronimo和Shagrithaya的最新工作，后者证明了此类码具有与随机线性码共享局部性质的能力。我们的工作通过子空间设计性质以统一方式获得了这一结论。此外，我们的方法在列表恢复参数方面取得了一些改进。