Tamo-Barg code[1] 

Description

A family of \(q\)-ary polynomial evaluation codes that are optimal LRCs and for which \(q\) is comparable to \(n\).

Each length-\(k\) message \(\mu\) is encoded into a string of values of its corresponding polynomial \(f_\mu\) at points \(\alpha\) in some size-\(n\) set \(A\). This polynomial can be written as \begin{align} f_{\mu}\left(x\right)=\sum_{i=1}^{r-1}\sum_{j=0}^{k/r-1}\mu_{ij}x^{i}g(x)^{j}~, \tag*{(1)}\end{align} where \(\mu\) has been reorganized into an \(r \times k/r\) matrix, and where \(g(x)\) is a degree-\(r+1\) polynomial that is constant on elements of certain partitions of \(A\) [1; Sec. III.A]. This construction assumes that \(r\) divides \(k\), but a simple extension to other parameters can be formulated.

Decoding

Polynomial evaluation over \(r\) points [1].

Parents

Child

References

[1]
I. Tamo and A. Barg, “A Family of Optimal Locally Recoverable Codes”, IEEE Transactions on Information Theory 60, 4661 (2014) arXiv:1311.3284 DOI
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Zoo Code ID: tamo_barg

Cite as:
“Tamo-Barg code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/tamo_barg
BibTeX:
@incollection{eczoo_tamo_barg, title={Tamo-Barg code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/tamo_barg} }
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Cite as:

“Tamo-Barg code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/tamo_barg

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/distributed_storage/lrc/tamo_barg.yml.