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.
Notes
See corresponding MinT database entry [1].
Parent
- Generalized RS (GRS) code — Extended GRS codes can be thought of as GRS codes that include an evaluation point of zero.
Cousins
- Hyperoval code — Columns of parity-check matrices of triply extended RS codes consist of points of a hyperoval [2; Prop. 17.5].
- Maximum distance separable (MDS) code — An MDS GRS code can be extended to an MDS code ([3], Thm. 5.3.4). Extended and doubly extended narrow-sense RS codes are MDS ([3], Thms. 5.3.2 and 5.3.4), and there is an equivalence between the two for odd prime \(q\) [4].
- Generalized RM (GRM) code — GRM codes for univariate polynomials (\(m=1\)) reduce to extended RS codes [5].
- Projective geometry code — Columns of parity-check matrices of doubly extended narrow-sense RS codes consist of points of a normal rational curve [6; Def. 14.2.6].
- Simplex code — \(S(2,k)\) is an extended RS code [7].
References
- [1]
- 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
- [2]
- J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
- [3]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [4]
- 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
- [5]
- J. B. Little, “Algebraic geometry codes from higher dimensional varieties”, (2008) arXiv:0802.2349
- [6]
- W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [7]
- 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
Page edit log
- Victor V. Albert (2022-07-19) — most recent
Cite as:
“Extended GRS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/extended_reed_solomon