Interleaved RS (IRS) code


A modification of RS codes where multiple polynomials are used to define each codeword. Each codeword \(\mu\) of a \(t\)-interleaved RS code is a string of values of the corresponding set \(\{f_\mu^{(1)},f_\mu^{(2)},\cdots,f_\mu^{(t)}\}\) of \(t\) polynomials at the points \(\alpha_i\). The vector codewords can be arranged in an array whose rows are ordinary RS codes for each polynomial \(f^{j}\), yielding the encoding \begin{align} \mu\to\left( \begin{split} f_{\mu}^{(1)}\left(\alpha_{1}\right) & f_{\mu}^{(1)}\left(\alpha_{2}\right) & \cdots & f_{\mu}^{(1)}\left(\alpha_{n}\right)\\ f_{\mu}^{(2)}\left(\alpha_{1}\right) & f_{\mu}^{(2)}\left(\alpha_{2}\right) & & f_{\mu}^{(2)}\left(\alpha_{n}\right)\\ \vdots & & \ddots & \vdots\\ f_{\mu}^{(t)}\left(\alpha_{1}\right) & f_{\mu}^{(t)}\left(\alpha_{2}\right) & \cdots & f_{\mu}^{(t)}\left(\alpha_{n}\right) \end{split}\right)~. \end{align}


Decoder that corrects up to \(1-\frac{2k+n}{3n}\) fraction of random errors [1].Decoder that corrects up to \(1-(\frac{k}{n})^{2/3}\) fraction of random errors [2].


The cross-interleaved RS (CIRC), an IRS code using two shortened RS codes and two forms of interleaving, was used for compact discs (CDs) [3] (see Ref. [4], Sec. 5.6 and Ref. [5], Ch. 4).



  • Parvaresh-Vardy (PV) code — PV codes are IRS codes with specific algebraic relations between the codeword polynomials that allow for efficient list decoding.
  • Reed-Solomon (RS) code — An IRS code utilizing one polynomial \(f\) reduces to an RS code.


D. Bleichenbacher, A. Kiayias, and M. Yung, “Decoding interleaved Reed–Solomon codes over noisy channels”, Theoretical Computer Science 379, 348 (2007). DOI
D. Coppersmith and M. Sudan, “Reconstructing curves in three (and higher) dimensional space from noisy data”, Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03 (2003). DOI
Odaka K., Sako Y., Iwamoto I., Doi T.; Vries L.B.; SONY: Error correctable data transmission method (Patent US4413340) filing date May 21, 1980.
W. C. Huffman and V. Pless, Fundamentals of Error-correcting Codes (Cambridge University Press, 2003). DOI
S. B. Wicker and V. K. Bhargava, Reed-solomon Codes and Their Applications (IEEE, 1999). DOI

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Internal code ID: interleaved_reed_solomon

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“Interleaved RS (IRS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_interleaved_reed_solomon, title={Interleaved RS (IRS) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Interleaved RS (IRS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.