Multi-Erasure Locally Recoverable Codes Over Small Fields: A Tensor Product Approach

• Published in: IEEE transactions on information theory Vol. 66; no. 5; pp. 2609 - 2624
• Main Authors: Huang, Pengfei, Yaakobi, Eitan, Siegel, Paul H.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-05-2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; capacity-achieving; compound channel; Compounds; Data loss; Decoding; Linear codes; Locally recoverable codes; Maintenance engineering; Mathematical analysis; Product codes; small fields; Storage systems; tensor product codes; Tensors; Upper bound
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2019.2962012

Erasure codes play an important role in storage systems to prevent data loss. In this work, we study a class of erasure codes called Multi-Erasure Locally Recoverable Codes (ME-LRCs) for storage arrays. Compared to previous related works, we focus on the construction of ME-LRCs over small fields. Our main contribution is a general construction of ME-LRCs based on generalized tensor product codes, and an analysis of their erasure-correcting properties. A decoding algorithm tailored for erasure recovery is given, and correctable erasure patterns are identified. We then prove that our construction yields optimal ME-LRCs with a wide range of code parameters, and present some explicit ME-LRCs over small fields. Next, we show that generalized integrated interleaving (GII) codes can be treated as a subclass of generalized tensor product codes, thus defining the exact relation between these codes. Finally, ME-LRCs are investigated in a probabilistic setting. We prove that ME-LRCs based upon a generalized tensor product construction can achieve the capacity of a compound erasure channel consisting of a family of erasure product channels.