# 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

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

## Child

## 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).
- \(q\)-ary linear LTC — Duals of BCH codes are locally testable [20].
- Justesen code — Using more general BCH codes instead of RS codes can improve the parameters of the Justesen codes [21].
- Reed-Solomon (RS) code — Narrow-sense RS codes are BCH codes [22; Remark 15.3.21][17; Thms. 5.2.1 and 5.2.3]. Their minimal distance is equal to their designed distance [23; 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 [24].
- Qubit BCH code — BCH codes are used to construct qubit BCH codes via the CSS and stabilizer-over-\(GF(4)\) constructions.
- Galois-qudit BCH code

## References

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- 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.
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- 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
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- A. Couvreur, H. Randriambololona, "Algebraic Geometry Codes and Some Applications." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
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- [24]
- A. Vardy and Y. Be’ery, “Bit-level soft-decision decoding of Reed-Solomon codes”, IEEE Transactions on Communications 39, 440 (1991) DOI

## Page edit log

- Victor V. Albert (2022-07-13) — most recent
- Muhammad Junaid Aftab (2022-04-21)

## 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