Description
A GRS code extended by one extra coordinate to form an \([n+1,k,n-k+2]_q\) MDS code. In projective language, this corresponds to adding one more evaluation point, often interpreted as the point at infinity; in suitable equivalent descriptions, one may instead use an affine point such as \(0\). 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 a projective-line 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
- Generalized RS (GRS) code— Extended GRS codes can be thought of as GRS codes that include an evaluation point of zero.
- 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].
- Narrow-sense RS code— A narrow-sense RS code can be extended once, twice, or three times.
- Reed-Solomon (RS) code— Extending an RS code by one evaluation point, often interpreted as the point at infinity, yields an extended RS code.
- 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 [6; Def. 14.2.6].
- \(q\)-ary simplex code— \(q\)-ary simplex codes for \(m=2\) can be thought of as extended RS codes [3].
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://web.archive.org/web/20240420202309/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]
- L. Storme, “Coding Theory and Galois Geometries.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [7]
- J. B. Little, “Algebraic geometry codes from higher dimensional varieties”, (2008) arXiv:0802.2349
- [8]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [9]
- F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
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