We develop a duality theory of locally recoverable codes (LRCs) and apply it to establish a series of new bounds on their parameters. We introduce and study a refined notion of weight distribution that captures the code's locality. Using a duality result analogous to a MacWilliams identity, we then derive an LP-type bound that improves on the best known bounds in several instances. Using a dual distance bound and the theory of generalized weights, we obtain non-existence results for optimal LRCs over small fields. In particular, we show that an optimal LRC must have both minimum distance and block length relatively small compared to the field size.

翻译：我们发展了局部可恢复码（LRCs）的对偶理论，并将其应用于建立一系列关于其参数的新界。我们引入并研究了捕捉码局部性的权重分布的精细概念。利用类似于MacWilliams恒等式的对偶结果，我们推导出线性规划型界，该界在若干实例中改进了已知最优界。通过使用对偶距离界和广义权重理论，我们得到了小域上最优LRCs的不存在性结果。特别地，我们证明了一个最优LRC必须具有相对域大小而言较小的最小距离和块长度。