Bobrova, Elisaveta (2025) Constructing optimal locally recoverable codes. Bachelor's Thesis, Mathematics.
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Abstract
This paper investigates constructions of optimal locally recoverable codes (LRC codes), a class of error-correcting codes that enable efficient recovery of erased codeword symbols using a small num- ber of other symbols and achieve a Singleton-type bound specific to LRC codes. The focus is on the polynomial evaluation framework introduced by Itzhak Tamo and Alexander Barg, which gen- eralizes Reed-Solomon codes by incorporating local recovery constraints into the encoding process. The central construction relies on evaluating structured encoding polynomials over finite fields at partitioned subsets of the field, where locality is ensured through the use of “good polynomials” — polynomials constant on each recovering set. Two algebraic methods for constructing good polyno- mials are developed: one based on annihilator polynomials of multiplicative or additive subgroups of finite fields, and another leveraging the interplay between additive and multiplicative structures of finite fields. In addition, we present a generalization of the original construction, and explore a systematic encoding procedure. The paper is complemented by detailed examples over small fields.
| Item Type: | Thesis (Bachelor's Thesis) |
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| Supervisor name: | Kilicer, P. and Salgado Guimaraes da Silva, C. |
| Degree programme: | Mathematics |
| Thesis type: | Bachelor's Thesis |
| Language: | English |
| Date Deposited: | 25 Jul 2025 06:56 |
| Last Modified: | 28 Jul 2025 07:51 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/36388 |
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