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 Title: On error correcting codes for distributed storage Author(s): Li, Xiao Director of Research: Duursma, Iwan Doctoral Committee Chair(s): Reznick, Bruce Doctoral Committee Member(s): Yong, Alexander; Milenkovic, Olgica Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Coding Theory, Distributed Storage Abstract: Two popular directions of error correcting codes for distributed storage are codes with additional recovery or regenerating properties. First we have codes for additional recovery properties. Codewords in array format find applications in disk storage where columns are stored on different disks in combination with parity checks across disks that protect data against disk failures. The addition of global parities protects against sector failures on any of the disks while keeping storage overhead low. We construct sector-disk array codes that tolerate any combination of two disk failures and three sector failures with minimal overhead. This constructs for the first time codes with these parameters without relying on exhaustive search. In the regenerating direction we have some modified layered codes in a two stage construction that gives regenerating codes with small field size. For more general parameters we define a Johnson graph code as a subspace of labelings of the vertices in a Johnson graph with the property that labelings are uniquely determined by their restriction to vertex neighborhoods specified by the parameters of the code. We give a construction and main properties for the codes. We show their role in the concatenation of layered codes to give regenerating codes for storage systems. Focusing on the Minimum Storage regenerating (MSR) point with $d=n-1$, we present graphical representations of codes with parameters \\ $((n,k,d), (\alpha, \beta)) = ((qt, q(t-1), qt-1),(q^t, q^{t-1}))$ over small field size. Issue Date: 2020-05-05 Type: Thesis URI: http://hdl.handle.net/2142/107954 Rights Information: Copyright 2020 Xiao Li Date Available in IDEALS: 2020-08-26 Date Deposited: 2020-05
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