Generalized RS (GRS) code 


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}


The code can detect \(n-k\) errors, and can correct errors \( \left\lfloor (n-k)/2\right\rfloor \) errors.


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 \(O(n^2)\) [13].Guruswami-Sudan list decoder [4] and modification by Koetter-Vardy for soft-decision decoding [5].


Commonly used in mass storage systems such as CDs, DVDs, QR codes etc.Various cloud storage systems [6].Public-key cryptosystems generalizing those that used Goppa codes [79], some of which were proven to be insecure [10]. More recent works focus on methods to mask the algebraic structure using subcodes of GRS codes [11]. For example, a key-recovery attack was developed in Ref. [12] for a variant of masking method proposed in Ref. [13].





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Zoo Code ID: generalized_reed_solomon

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“Generalized RS (GRS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
  title={Generalized RS (GRS) code},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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“Generalized RS (GRS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.