Narrow-sense RS code

Narrow-sense RS code [1–3]

Description

An $[q-1,k,n-k+1]_q$ RS code whose points $\alpha_i$ are all $(i-1)$st powers of a primitive element $\alpha$ of $\mathbb{F}_q$.

A narrow-sense RS code encodes a message $\mu=\{\mu_0,\cdots,\mu_{k-1}\}$ into $\{f_\mu(1),f_\mu(\alpha),\cdots,f_\mu(\alpha^{n-1})\}$ 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 $\alpha^{i-1}$, \begin{align} \mu\to\left( f_{\mu}\left(1\right),f_{\mu}\left(\alpha\right),\cdots,f_{\mu}\left(\alpha^{n-1}\right)\right) \,.\tag*{$(2)$} \end{align}

In an alternative convention (not used here), this code is called an RS code, and the general-root case is a generalized RS code.

Cousins

Extended GRS code— A narrow-sense RS code can be extended once, twice, or three times.
Maximum distance separable (MDS) code— Extended and doubly extended narrow-sense RS codes are MDS [4; Thms. 5.3.2 and 5.3.4][5; Ch. 11], and there is an equivalence between the two for odd prime $q$ [6].
Projective geometry code— Columns of parity-check matrices of doubly extended narrow-sense RS codes consist of points of a normal rational curve [7; Def. 14.2.6].

Member of code lists

Primary Hierarchy

Classical Domain

Galois-field Kingdom

$q$-ary codeECC

Additive $q$-ary codeECC

Linear $q$-ary codeECC

Evaluation codeAG ECC

Evaluation AG code

Residue AG code

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

Reed-Solomon (RS) codeEvaluation AG Linear $q$-ary LRC Distributed-storage ECC

Parents

Reed-Solomon (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 $\mathbb{F}_q$.

Bose–Chaudhuri–Hocquenghem (BCH) codeLinear $q$-ary Cyclic Constacyclic Skew-cyclic ECC

Narrow-sense RS codes are $q$-ary BCH codes [8; Remark 15.3.21][4; Thms. 5.2.1 and 5.2.3]. Their minimal distance is equal to their designed distance [9; pg. 81].

Narrow-sense RS code

References

[1]
K. A. Bush, “Orthogonal Arrays of Index Unity”, The Annals of Mathematical Statistics 23(3), 426–34 (1952)
[2]
K. A. Bush, “Orthogonal Arrays of Index Unity”, The Annals of Mathematical Statistics 23, 426 (1952) DOI
[3]
I. S. Reed and G. Solomon, “Polynomial Codes Over Certain Finite Fields”, Journal of the Society for Industrial and Applied Mathematics 8, 300 (1960) DOI
[4]
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
[5]
R. Roth, Introduction to Coding Theory (Cambridge University Press, 2006) DOI
[6]
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
[7]
L. Storme, “Coding Theory and Galois Geometries”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 285-310 DOI
[8]
A. Couvreur, H. Randriambololona, “Algebraic Geometry Codes and Some Applications”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 311-362 DOI
[9]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI

Page edit log

Victor V. Albert (2026-06-08) — most recent
Victor V. Albert (2024-08-13)

Cite as:

“Narrow-sense RS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/narrow_sense_reed_solomon, arXiv:2606.11484

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