## Description

An \([n,k,n-k+1]_q\) linear code based on polynomials over \(GF(q)\).

Let \(\{\alpha_1,\cdots,\alpha_n\}\) be \(n\) distinct points in \(GF(q)\). An RS code encodes a message \(\mu=\{\mu_0,\cdots,\mu_{k-1}\}\) into \(\{f_\mu(\alpha_1),\cdots,f_\mu(\alpha_n)\}\) using a message-dependent polynomial \begin{align} f_\mu(x)=\mu_0+\mu_1 x + \cdots + \mu_{k-1}x^{k-1}. \tag*{(1)}\end{align} In other words, each message \(\mu\) is encoded into a string of values of the corresponding polynomial \(f_\mu\) at the points \(\alpha_i\), \begin{align} \mu\to\left( f_{\mu}\left(\alpha_{1}\right),f_{\mu}\left(\alpha_{2}\right),\cdots,f_{\mu}\left(\alpha_{n}\right)\right) \,. \tag*{(2)}\end{align}

An RS code with length \(n=q-1\) whose points \(\alpha_i\) are \(i-1\)st powers of a primitive \(n\)th root of unity is a narrow-sense RS code. In an alternative convention (not used here), the primitive-root case is called an RS code, and the general-root case is a generalized RS code.

## Protection

## Encoding

## Decoding

## Realizations

## Notes

## Parents

- Generalized RS (GRS) code — A GRS code for which all multipliers \(v_i\) are unity reduces to an RS code.
- Interleaved RS (IRS) code — An IRS code utilizing one polynomial \(f\) reduces to an RS code.
- Folded RS (FRS) code — An FRS code with no extra grouping (\(m=1\)) reduces to an RS code.
- Multiplicity code — Univariate multiplicity codes of degree \(s=1\) reduce to RS codes.
- Tamo-Barg code — Tamo-Barg codes reduce to RS codes when \(r=k\).
- DNA storage code — RS codes have been used for DNA storage [34].

## Child

- \(q\)-ary parity-check code — RS codes for \(k=n-1\) are parity-check codes [37].

## Cousins

- Maximum distance separable (MDS) code — RS codes have distance \(n-k+1\), saturating the Singleton bound. If \(k \leq p\), then all linear MDS codes \( [n,k,n-k+1]_{p^m} \) are RS codes [38].
- Bose–Chaudhuri–Hocquenghem (BCH) code — Narrow-sense RS codes are BCH codes [39; Remark 15.3.21][40; Thms. 5.2.1 and 5.2.3]. Their minimal distance is equal to their designed distance [41; pg. 81]. Moreover, an RS code can be represented as a union of cosets, with each coset being an interleaver of several binary BCH codes [23].
- Cyclic linear \(q\)-ary code — If the length divides \(q-1\), then it is possible to construct a cyclic RS code. Such codes a Abelian group-algebra codes [42; Example 16.4.9].
- Tensor-product code — Tensor codes constructed from RS codes are robustly testable [43].
- \(q\)-ary linear LTC — RS codes can be used to construct LTCs encoding \(k\) bits with length \(k \text{polylog}(k)\) and query complexity \(\text{polylog}(k)\) [44].
- Justesen code — An RS code is the outer code of Justesen codes.
- Rank-modulation Gray code (RMGC) — RS codes can be used to design rank modulation codes [45].
- B-code — B-codes can be interpreted as RS codes over polynomials whose symbols lie in Galois rings [46,47].
- Maximum-rank distance (MRD) code — MRD rank-metric codes can be thought of as matrix analogues of MDS Reed-Solomon codes as both constructions utilize a Vandermonde matrix [48].
- Convolutional code — Convolutional codes are often used in concatenation with Reed-Solomon codes for communication [49].
- Glynn code — The only other inequivalent \([10,5,6]_9\) code is a Reed-Solomon code, which is the unique Euclidean self-dual code for its parmaters, and which is not Hermitian self-dual [50–52].
- Hexacode — The dual of the shortened hexacode code is a \([5,2,4]_4\) doubly extended RS code [53; Exam. A].
- Subsystem qubit stabilizer code — Primitive RS codes yield subsystem stabilizer codes via the subsystem extension of the Hermitian construction to subsystem codes [54; Exam. 3].
- Approximate secret-sharing code — The classical information in this code is encoded using a Reed-Solomon code.
- Galois-qudit RS code — Galois-qudit RS codes are CSS codes constructed from Reed-Solomon codes.

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## Page edit log

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- Victor V. Albert (2022-04-28)
- Mustafa Doger (2022-04-03)
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## Cite as:

“Reed-Solomon (RS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/reed_solomon