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Codes over rings, especially over Galois rings, have been extensively studied for nearly three decades due to their similarity to linear codes over finite fields. A distributed storage system uses a linear code to encode a large file across several nodes. If one of the nodes fails, a linear exact repair scheme efficiently recovers the failed node by accessing and downloading data from the rest of the servers of the storage system. In this paper, we develop a linear repair scheme for free maximum distance separable codes, which coincide with free maximum distance with respect to the rank codes over Galois rings. In particular, we give a linear repair scheme for full-length Reed–Solomon codes over a Galois ring.
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Number theoretical locally recoverable codes
In this paper, we give constructions for infinite sequences of finite nonlinear locally recoverable codes [Formula: see text] over a product of finite fields arising from basis expansions in algebraic number fields. The codes in our sequences have increasing length and size, constant rate, fixed locality, and minimum distance going to infinity.
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- Award ID(s):
- 2127742
- PAR ID:
- 10508686
- Publisher / Repository:
- World Scientific
- Date Published:
- Journal Name:
- Journal of Algebra and Its Applications
- ISSN:
- 0219-4988
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1142/S0219498825500604
