Roth-Lempel code[1] 

Description

Member of a \(q\)-ary linear code family that includes many examples of MDS codes that are not GRS codes.

The generator matrix of a Roth-Lempel code is \begin{align} G=\left(\begin{array}{cccccc} \alpha_{1}^{k-1} & \alpha_{2}^{k-1} & \cdots & \alpha_{n}^{k-1} & 1 & 0\\ \alpha_{1}^{k-2} & \alpha_{2}^{k-2} & \cdots & \alpha_{n}^{k-2} & 0 & 1\\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots\\ \alpha_{1}^{2} & \alpha_{2}^{2} & \cdots & \alpha_{n}^{2} & 0 & 0\\ \alpha_{1} & \alpha_{2} & \cdots & \alpha_{n} & 0 & 0\\ 1 & 1 & \cdots & 1 & 0 & 0 \end{array}\right)~, \tag*{(1)}\end{align} where \(\{\alpha_j\}\) is a set of elements of \(GF(q)\). The code is MDS if no subset of \(k-1\) elements sums to zero.

Parents

Cousin

References

[1]
R. M. Roth and A. Lempel, “A construction of non-Reed-Solomon type MDS codes”, IEEE Transactions on Information Theory 35, 655 (1989) DOI
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Zoo Code ID: roth_lempel

Cite as:
“Roth-Lempel code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/roth_lempel
BibTeX:
@incollection{eczoo_roth_lempel, title={Roth-Lempel code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/roth_lempel} }
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Cite as:

“Roth-Lempel code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/roth_lempel

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