# Bose–Chaudhuri–Hocquenghem (BCH) code[1]

## Description

Cyclic \(q\)-ary code, with \(n\) and \(q\) relatively coprime, whose zeroes are consecutive powers of a primitive \(n\)th root of unity \(\alpha\). More precisely, the generator polynomial of a BCH code of designed distance \(\delta\geq 1\) is the lowest-degree monic polynomial with zeroes \(\{\alpha^b,\alpha^{b+1},\cdots,\alpha^{b+\delta-2}\}\) for some \(b\geq 0\). BCH codes are called narrow-sense when \(b=1\), and are called primitive when \(n=q^r-1\) for some \(r\geq 2\).

The code dimension is related to the multiplicative order of \(q\) modulo \(n\), i.e., the smallest integer \(m\) such that \(n\) divides \(q^m-1\). The dimension of a BCH code is at least \(n-m(\delta-1)\). The field \(GF(q^m)\) is the smallest field containing the above root of unity \(\alpha\), and is the splitting field of the polynomial \(x^n-1\) (see Cyclic-to-polynomial correspondence).

## Protection

## Rate

## Decoding

## Realizations

## Notes

## Parents

- Generalized RS (GRS) code — BCH codes are subfield subcodes of GRS codes.
- Cyclic linear \(q\)-ary code

## Cousins

- Classical Goppa code — Narrow-sense BCH codes are Goppa codes with \(L=\{1,\alpha^{-1},\cdots,\alpha^{1-n}\}\) and \(G(x)=x^{\delta-1}\) ([17], pg. 522).
- Binary BCH code
- Galois-qudit BCH code
- Qubit BCH code — BCH codes are used to construct qubit BCH codes via the CSS and stabilizer-over-\(GF(4)\) constructions.
- Reed-Solomon (RS) code — Narrow-sense RS codes are BCH codes ([20], Remark 15.3.21; [17], Thms. 5.2.1 and 5.2.3). Moreover, an RS code can be represented as a union of cosets, with each coset being an interleaver of several binary BCH codes [21].

## References

- [1]
- D. Gorenstein and N. Zierler, “A Class of Error-Correcting Codes in $p^m $ Symbols”, Journal of the Society for Industrial and Applied Mathematics 9, 207 (1961). DOI
- [2]
- F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
- [3]
- E. Berlekamp, “Nonbinary BCH decoding (Abstr.)”, IEEE Transactions on Information Theory 14, 242 (1968). DOI
- [4]
- J. Massey, “Shift-register synthesis and BCH decoding”, IEEE Transactions on Information Theory 15, 122 (1969). DOI
- [5]
- E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, 1968
- [6]
- H. Burton, “Inversionless decoding of binary BCH codes”, IEEE Transactions on Information Theory 17, 464 (1971). DOI
- [7]
- W. W. Peterson and E. J. Weldon, Error-correcting codes. MIT press 1972.
- [8]
- R. Gallager, Information Theory and Reliable Communication (Springer Vienna, 1972). DOI
- [9]
- W. Peterson, “Encoding and error-correction procedures for the Bose-Chaudhuri codes”, IEEE Transactions on Information Theory 6, 459 (1960). DOI
- [10]
- R.E. Blahut, Theory and practice of error-control codes, Addison-Wesley 1983.
- [11]
- Y. Sugiyama et al., “A method for solving key equation for decoding goppa codes”, Information and Control 27, 87 (1975). DOI
- [12]
- R. McEliece, The Theory of Information and Coding (Cambridge University Press, 2002). DOI
- [13]
- V. Guruswami and M. Sudan, “Improved decoding of Reed-Solomon and algebraic-geometric codes”, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280). DOI
- [14]
- R. Koetter and A. Vardy, “Algebraic soft-decision decoding of reed-solomon codes”, IEEE Transactions on Information Theory 49, 2809 (2003). DOI
- [15]
- S. Zhu, Z. Sun, and X. Kai, “A Class of Narrow-Sense BCH Codes”, IEEE Transactions on Information Theory 65, 4699 (2019). DOI
- [16]
- S. Lin and D. J. Costello, Error Control Coding, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 2004.
- [17]
- W. C. Huffman and V. Pless, Fundamentals of Error-correcting Codes (Cambridge University Press, 2003). DOI
- [18]
- Michael Helmling, Stefan Scholl, Florian Gensheimer, Tobias Dietz, Kira Kraft, Stefan Ruzika, and Norbert Wehn. Database of Channel Codes and ML Simulation Results. URL, 2022.
- [19]
- Rudolf Schürer and Wolfgang Ch. Schmid. “Cyclic Codes (BCH-Bound).” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CCyclic-BCHBound.html
- [20]
- W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021). DOI
- [21]
- A. Vardy and Y. Be'ery, “Bit-level soft-decision decoding of Reed-Solomon codes”, IEEE Transactions on Communications 39, 440 (1991). DOI

## Zoo code information

## Cite as:

“Bose–Chaudhuri–Hocquenghem (BCH) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_bch