Description
An \([n,k,n-k+1]_q\) linear code that is a modification of the RS code where codeword polynomials are multiplied by additional prefactors.
Each message \(\mu\) is encoded into a string of values of the corresponding polynomial \(f_\mu\) at the points \(\alpha_i\), multiplied by a corresponding nonzero factor \(v_i \in GF(q)\), \begin{align} \mu\to\left( v_{1}f_{\mu}\left(\alpha_{1}\right),v_{2}f_{\mu}\left(\alpha_{2}\right),\cdots,v_{n}f_{\mu}\left(\alpha_{n}\right)\right)~. \tag*{(1)}\end{align}
Protection
The code can detect \(n-k\) errors, and can correct errors \( \left\lfloor (n-k)/2\right\rfloor \) errors.Decoding
The decoding process of GRS codes reduces to the solution of a polynomial congruence equation, usually referred to as the key equation. Decoding schemes are based on applications of the Euclid algorithm to solve the key equation.Berlekamp-Massey decoder with runtime of order \(O(n^2)\) [2–4].Guruswami-Sudan list decoder [5,6] and modification by Koetter-Vardy for soft-decision decoding [7].Hard-decision decoder for errors within the Singleton bound [8].Realizations
Commonly used in mass storage systems such as CDs, DVDs, QR codes etc.Various cloud storage systems [9].A variation of the McEliece public-key cryptosystem [10,11] by Niederreiter [12] replaced the generator matrix by the parity check matrix of a GRS code. This was proven to be insecure since the public key exposes the algebraic structure of code [13]. More recent works focus on methods to mask the algebraic structure using subcodes of GRS codes [14]. For example, a key-recovery attack was developed in Ref. [15] for a variant of masking method proposed in Ref. [16].Cousins
- Concatenated code— Concatenations of GRS codes with random linear codes almost surely attain the GV bound [1].
- Random code— Concatenations of GRS codes with random linear codes almost surely attain the GV bound [1].
- Linear \(q\)-ary code— Concatenations of GRS codes with random linear codes almost surely attain the GV bound [1].
- Hermitian code— Hermitian codes are concatenated GRS codes [17].
- Goppa code— Goppa codes are \(GF(q)\)-subfield subcode of the dual of the GRS code over \(GF(q^m)\) with evaluation points \(\alpha_i\) and factors \(v_i=G(\alpha_i)^{-1}\) ([18], pg. 523; [19]).
- Alternant code— Alternant codes are subfield subcodes of GRS codes [20].
- Quantum maximum-distance-separable (MDS) code— Some quantum MDS codes are constructed from cyclic and constacyclic codes [21] which are GRS codes [22,23].
- Folded quantum RS (FQRS) code— A folded quantum generalized RS (GRS) code can be constructed in similar fashion from GRS codes as FQRS codes are constructed from FRS codes [24; Sec. 3].
- Galois-qudit GRS code
Member of code lists
Primary Hierarchy
Parents
GRS (RS) codes are in one-to-one correspondence with evaluation AG codes of univariate polynomials \(f\) with \(\cal X\) being the projective (affine) line [19,27][25; Thm. 15.3.24][26; Ch. 3.2].
Maximum distance separable (MDS) codeLinear \(q\)-ary OA LRC Distributed-storage \(t\)-design Universally optimal ECC
GRS codes have distance \(n-k+1\), saturating the Singleton bound.
GRS codes are used in various cloud storage systems [9].
Generalized RS (GRS) code
Children
Extended GRS codes can be thought of as GRS codes that include an evaluation point of zero.
A GRS code for which all multipliers \(v_i\) are unity reduces to an RS code.
GRS codes can be used to construct quantum convolutional codes [28].
References
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- E. Berlekamp, “Nonbinary BCH decoding (Abstr.)”, IEEE Transactions on Information Theory 14, 242 (1968) DOI
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- J. Massey, “Shift-register synthesis and BCH decoding”, IEEE Transactions on Information Theory 15, 122 (1969) DOI
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- E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, 1968
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- V. Guruswami and M. Sudan, “Improved decoding of Reed-Solomon and algebraic-geometry codes”, IEEE Transactions on Information Theory 45, 1757 (1999) DOI
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- V. Guruswami and M. Sudan, “Improved decoding of Reed-Solomon and algebraic-geometric codes”, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280) 28 DOI
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Page edit log
- Victor V. Albert (2022-09-16) — most recent
- Victor V. Albert (2022-05-17)
- Muhammad Junaid Aftab (2022-04-21)
- Qingfeng (Kee) Wang (2021-12-20)
Cite as:
“Generalized RS (GRS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/generalized_reed_solomon