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. There also exist triply extended RS codes with parameters \([q+2,3,q-1]_q\) or \([q+2,q-1,4]_q\) [1].
Their automorphism groups have been identified [2].
Notes
See corresponding MinT database entry [3].Cousins
- Hyperoval code— Columns of parity-check matrices of triply extended RS codes consist of points of a hyperoval [1; Prop. 17.5].
- Maximum distance separable (MDS) code— A GRS code can be extended to an MDS code ([4], Thm. 5.3.4). Extended and doubly extended narrow-sense RS codes are MDS ([4], Thms. 5.3.2 and 5.3.4), and there is an equivalence between the two for odd prime \(q\) [5].
- \([2^m-1,m,2^{m-1}]\) simplex code— Simplex codes are extended RS codes [6].
- Narrow-sense RS code— A narrow-sense RS code can be extended once, twice, or three times.
- Roth-Lempel code— Roth-Lempel codes are doubly extended RS codes.
- Projective geometry code— Columns of parity-check matrices of doubly extended narrow-sense RS codes consist of points of a normal rational curve [7; Def. 14.2.6].
Primary Hierarchy
References
- [1]
- J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
- [2]
- A. Dür, “The automorphism groups of Reed-Solomon codes”, Journal of Combinatorial Theory, Series A 44, 69 (1987) DOI
- [3]
- 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. https://mint.sbg.ac.at/desc_CReedSolomon-extended.html
- [4]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [5]
- 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 14, 733 (2012) DOI
- [6]
- Rudolf Schürer and Wolfgang Ch. Schmid. “Simplex Code.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. https://mint.sbg.ac.at/desc_CSimplex.html
- [7]
- L. Storme, "Coding Theory and Galois Geometries." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [8]
- J. B. Little, “Algebraic geometry codes from higher dimensional varieties”, (2008) arXiv:0802.2349
- [9]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
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