Linear codes over finite extension fields have widespread applications in theory and practice. In some scenarios, the decoder has a sequential access to the codeword symbols, giving rise to a hierarchical erasure structure. In this paper we develop a mathematical framework for hierarchical erasures over extension fields, provide several bounds and constructions, and discuss potential applications in distributed storage and flash memories. Our results show intimate connection to Universally Decodable Matrices, as well as to Reed-Solomon and Gabidulin codes.

中文翻译：

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线性码的分层擦除校正

有限扩展域上的线性代码在理论和实践中都有广泛的应用。在某些情况下，解码器可以顺序访问码字符号，从而产生分层擦除结构。在本文中，我们为扩展字段上的分层擦除开发了一个数学框架，提供了一些边界和构造，并讨论了在分布式存储和闪存中的潜在应用。我们的结果表明与通用可腐烂矩阵以及里德-所罗门和伽比杜林编码紧密相关。