Extended GRS code 

Description

A GRS code with an additional parity-check coordinate with corresponding evaluation point of zero. In other words, an \([n+1,k,n-k+2]_q\) GRS code whose polynomials are evaluated at the points \((\alpha_1,\cdots,\alpha_n,0)\). The case when \(n=q-1\), multipliers \(v_i=1\), and \(\alpha_i\) are \(i-1\)st powers of a primitive \(n\)th root of unity is an extended narrow-sense RS code.

An \([q-1,k,q-k]_q\) narrow-sense RS code can be extended twice by adding two evaluation points (of which one can be zero) to yield a \([q+1,k,q-k+2]_q\) doubly extended narrow-sense RS code. The two extra columns sometimes correspond to evaluating at zero and infinity if one switches to projective coordinates, in which case the doubly extended GRS code is an evaluation code.

Their automorphism groups have been identified [1].

Notes

See corresponding MinT database entry [2].

Parent

Children

Cousins

References

[1]
A. Dür, “The automorphism groups of Reed-Solomon codes”, Journal of Combinatorial Theory, Series A 44, 69 (1987) DOI
[2]
Rudolf Schürer and Wolfgang Ch. Schmid. “Extended Reed–Solomon Code.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CReedSolomon-extended.html
[3]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[4]
J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
[5]
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
[6]
S. Ball, “On sets of vectors of a finite vector space in which every subset of basis size is a basis”, Journal of the European Mathematical Society 733 (2012) DOI
[7]
J. B. Little, “Algebraic geometry codes from higher dimensional varieties”, (2008) arXiv:0802.2349
[8]
Rudolf Schürer and Wolfgang Ch. Schmid. “Simplex Code.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CSimplex.html
[9]
L. Storme, "Coding Theory and Galois Geometries." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
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Zoo Code ID: extended_reed_solomon

Cite as:
“Extended GRS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/extended_reed_solomon
BibTeX:
@incollection{eczoo_extended_reed_solomon, title={Extended GRS code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/extended_reed_solomon} }
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Cite as:

“Extended GRS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/extended_reed_solomon

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/ag/rs/extended_reed_solomon.yml.