Reed-Solomon (RS) code

Reed-Solomon (RS) code[1–3]

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}

Protection

Since each polynomial \(f_{\mu}\) is of degree less than \(k\), it has less than \(k\) roots; this is called the degree mantra. Therefore, the polynomial can be determined from its values at \(k\) points. This means that RS codes can correct erasures on up to \(n-k\) registers. The resulting distance, \(d=n-k+1\), saturates the Singleton bound.

Encoding

Bit-serial encoder [4].\([n,k,n-k+1]\) RS code requires an order \(O(n^2)\) operations while encoding if a straightforward matrix multiplication is employed and \(k=O(n)\). Using the FFT algorithm, complexity of evaluating a polynomial at \(n\) roots of unity becomes \(O(n\log n)\). The FFT can be generalized to finite fields and rings, which is referred as Number-theoretic Transform (NTT). However, for some values of \(n\), which can not be factorized into small primes or do not have \(n\) roots of unity, the FFT algorithm fails. Independently developed by [5,6] and generalized in Ref. [7], the additive FFT solves this problem by evaluating the polynomial at \(n-1\) roots of unity when \(n\) is power of 2.

Decoding

Decoding general RS codes is \(NP\)-hard [8].Although using iFFT has its counterpart iNNT for finite fields, the decoding is usually standard polynomial interpolation in \(k=O(n\log^2 n)\). However, in erasure decoding, encoded values are only erased in \(r\) points, which is a specific case of polynomial interpolation and can be done in \(O(n\log n)\) by computing product of the received polynomial and an erasure locator polynomial and using long division to find an original polynomial. The long division step can be omitted to increase speed further by only dividing the derivative of the product polynomial, and derivative of erasure locator polynomial evaluated at erasure locations.Berlekamp-Massey decoder with runtime of order \(O(n^2)\) [9,10].Gorenstein-Peterson-Zierler decoder with runtime of order \(O(n^3)\) [11,12] (see exposition in Ref. [13]).Berlekamp-Welch decoder with runtime of order \(O(n^3)\) [14] (see exposition in Ref. [15]), assuming that \(t \geq (n+k)/2\).Gao decoder using extended Euclidean algorithm [16].Fast-Fourier-transform decoder with runtime of order \(O(n \text{polylog}n)\) [17].List decoders try to find a low-degree bivariate polynomial \(Q(x,y)\) such that evaluation of \(Q\) at \((\alpha_i,y_i)\) is zero. By choosing proper degrees, it can be shown such polynomial exists by drawing an analogy between evaluation of \(Q(\alpha_i,y_i)\) and solving a homogenous linear equation (interpolation). Once this is done, one lists roots of \(y\) that agree at \(\geq t\) points. The breakthrough Sudan list-decoding algorithm corrects up to \(1-\sqrt{2R}\) fraction of errors asymptotically in \(n\) [18]. Roth and Ruckenstein proposed a modified key equation that allows for correction of more than \(\left\lfloor (n-k)/2 \right\rfloor\) errors [19]. The Guruswami-Sudan algorithm improved the Sudan algorithm to \(1-\sqrt{R}\) [20], meaning that RS codes achieve list-decoding capacity; see Ref. [21] for bounds. It was later shown that generic RS codes achieve list-decoding capacity [22]. A modification of the Guruswami-Sudan algorithm by Koetter and Vardy is used for soft-decision decoding [23] (see also Ref. [24]). Subcodes of RS codes whose evaluation points lie in a subfield can be decoded up to the \(1-R\) [25]. List decoding of RS codes is known as noisy polynomial interpolation in cryptography [26].The ubiquity of RS codes has yielded off-the-shelf VLSI intergrated-circuit decoding hardware [27] (see also Ref. [28], Ch. 5 and 10).

Realizations

RS Product Code (RSPC) was used in DVDs (see Ref. [28], Ch. 4).DSL technologies and their variants against impluse noise [29].Cryptographic primitives based on the hardness of decoding RS codes for more than \(1-\sqrt{k/n}+\epsilon\) errors. This is equivalent to the polynomial reconstruction problem [30].RS codes as outer codes concatenated with convolutional codes are used indirectly in space exploration programs such as Voyager and Galileo. RS codes were part of a telemetry channel coding standard issued by the Consultative Committee for Space Data Systems (see Ref. [28], Ch. 3).Automatic repeat request (ARQ) data transmission protocols (see Ref. [28], Ch. 7).Slow-frequency-hop spread-spectrum transmission (see Ref. [28], Chs. 8-9).RS codes over \(q=2^m\) are used in RAID 6 [31,32]; see [33].Coded sharding designs in blockchains to increase efficiency [34].Used in QR-Codes to retrieve damaged barcodes [35].Wireless communication systems such as 3G, DVB, and WiMAX [36].Correcting pooled testing results for SARS-CoV-2 [37].DNA storage [38].

Notes

See Kaiserslautern database [39] for explicit codes.See corresponding MinT database entry [40].Popular summary in Quanta Magazine.Certain structured classical optimization problems can be mapped into decoding and list decoding RS codes via the Decoded Quantum Interferomentry (DQI) algorithm [41,42].

Cousins

DNA storage code— RS codes have been used for DNA storage [38].
Maximum distance separable (MDS) code— If \(k \leq p\), then all linear MDS codes \( [n,k,n-k+1]_{p^m} \) are RS codes [43].
Bose–Chaudhuri–Hocquenghem (BCH) code— An RS code can be represented as a union of cosets, with each coset being an interleaver of several binary BCH codes [24]. BCH codes are subfield subcodes of RS codes.
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 [44; Example 16.4.9].
Tensor-product code— Tensor codes constructed from RS codes are robustly testable [45].
\(q\)-ary linear LTC— RS codes can be used to construct LTCs encoding \(k\) bits with length \(k \text{polylog}(k)\) and query complexity \(\text{polylog}(k)\) [46].
Private information retrieval (PIR) code— RS codes can be used to construct PIR codes [47].
\([8,4,4]\) extended Hamming code— The \([4,2,3]_4\) RS code is the smallest example of a quaternary quadratic-residue code and can be mapped to the \([8,4,4]\) extended Hamming code [48; Sec. 2.4.2] by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) [49].
Quadratic-residue (QR) code— The \([4,2,3]_4\) RS code is the smallest example of a quaternary quadratic-residue code and can be mapped to the \([8,4,4]\) extended Hamming code [48; Sec. 2.4.2] by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) [49].
Analog RS code— Analog RS codes are versions of RS codes over the real and complex numbers.
Justesen code— An RS code is the outer code of Justesen codes.
Array code— RS codes over \(q=2^m\) are used in RAID 6 [31,32]; see [33].
B-code— B-codes can be interpreted as RS codes over polynomials whose symbols lie in Galois rings [50,51].
Maximum-rank distance (MRD) code— MRD rank-metric codes can be thought of as matrix analogues of MDS RS codes as both constructions utilize a Vandermonde matrix [52].
Linearized RS code— RS codes are particular cases of linearized RS codes because the sum-rank metric generalizes the Hamming metric [53].
Hyperbolic evaluation code— Hyperbolic evaluation codes were initially formulated as generalized concatenations (a.k.a. cascades) of RS codes [54,55].
Convolutional code— Convolutional codes can be constructed from [56] and concatenated with [57] RS codes.
Pyramid code— A pyramid code is an LRC whose generator matrix is that of an RS code in standard form, but some of whose columns are split into multiple columns
Glynn code— The only other inequivalent \([10,5,6]_9\) code is an RS code, which is the unique Euclidean self-dual code for its parameters, and which is not Hermitian self-dual [58–60].
Hexacode— The dual of the shortened hexacode code is a \([5,2,4]_4\) doubly extended RS code [61; Exam. A].
Berlekamp code— Berlekamp codes are obtained by first constructing an RS-like parity-check matrix out of a certain field extension of \(GF(p)\) and then taking the subfield subcode of the corresponding code; see [62; Ch. 10.6].
Subsystem qubit stabilizer code— Primitive RS codes yield subsystem stabilizer codes via the subsystem extension of the Hermitian construction to subsystem codes [63; Exam. 3].
Approximate secret-sharing code— The classical information in this code is encoded using an RS code.
Galois-qudit RS code— Galois-qudit RS codes are CSS codes constructed from RS codes.
Galois-qudit expander code— Hypergraph products of expander codes with RS inner codes yield \([[n,k\geq n^{1-\epsilon},d\geq n^{1/r}/\text{poly}(\log n)]]_q\) QLDPC Galois-qudit quantum expander codes with transversal \(C^{r-1} Z\) gates [64]. Balanced products of expander codes with RS inner codes yield \([q^{\text{polylog}(q)},k\geq n^{1-\epsilon},n/\text{poly}(\log n)]_q\) LTCs exhibiting the multiplication property [64].

Member of code lists

Primary Hierarchy

Classical Domain

Galois-field Kingdom

\(q\)-ary codeECC

Additive \(q\)-ary codeECC

Linear \(q\)-ary codeECC

Evaluation code

Evaluation AG codeAG ECC

Generalized RS (GRS) codeMDS Linear \(q\)-ary OA \(t\)-design Universally optimal LRC Distributed-storage ECC

Parents

Generalized RS (GRS) code

A GRS code for which all multipliers \(v_i\) are unity reduces to an RS code.

Interleaved RS (IRS) codeLinear \(q\)-ary ECC

An IRS code utilizing one polynomial \(f\) reduces to an RS code.

Folded RS (FRS) codeLinear \(q\)-ary ECC

An FRS code with no extra grouping (\(m=1\)) reduces to an RS code.

Multiplicity codeLinear \(q\)-ary ECC

Univariate multiplicity codes of degree \(s=1\) reduce to RS codes.

Tamo-Barg codeEvaluation Linear \(q\)-ary AG LRC Distributed-storage ECC

Tamo-Barg codes reduce to RS codes when \(r=k\).

RS NRT codeECC

RS NRT codes reduce to RS codes when the NRT metric is equivalent to the Hamming metric [65].

Reed-Solomon (RS) code

Children

Narrow-sense RS code

A narrow-sense RS is an RS code with length \(n=q-1\) whose points \(\alpha_i\) are all \((i-1)\)st powers of a primitive element of \(GF(q)\).

\(q\)-ary parity-check code

RS codes for \(k=n-1\) are parity-check codes [66].

References

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Victor V. Albert (2022-04-28)
Mustafa Doger (2022-04-03)
Victor V. Albert (2021-10-29)

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/ag/reed_solomon/rs/reed_solomon.yml.