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


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).


By the BCH bound, BCH code with designed distance \(\delta\) has true minimum distance \(d\geq\delta\). BCH codes with different designed distances may coincide, and the largest possible designed distance for a given code is the Bose distance; the true distance may still be larger than that.


Primitive BCH codes are asymptotically bad [2; pg. 269].


Berlekamp-Massey decoder with runtime of order \(O(n^2)\) [35] and modification by Burton [6]; see also [7,8].Gorenstein-Peterson-Zierler decoder with runtime of order \(O(n^3)\) [1,9] (see exposition in Ref. [10]).Sugiyama et al. modification of the extended Euclidean algorithm [11,12].Guruswami-Sudan list decoder [13] and modification by Koetter-Vardy for soft-decision decoding [14].


DVDs, disk drives, and two-dimensional bar codes [15].


See books [2,16,17] for expositions on BCH codes and code tables.See Kaiserslautern database [18] for explicit codes.See corresponding MinT database entry [19].





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E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, 1968
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R. Koetter and A. Vardy, “Algebraic soft-decision decoding of reed-solomon codes”, IEEE Transactions on Information Theory 49, 2809 (2003) DOI
S. Zhu, Z. Sun, and X. Kai, “A Class of Narrow-Sense BCH Codes”, IEEE Transactions on Information Theory 65, 4699 (2019) DOI
S. Lin and D. J. Costello, Error Control Coding, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 2004.
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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.
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.
T. Kaufman and S. Litsyn, “Almost Orthogonal Linear Codes are Locally Testable”, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) DOI
E. J. Weldon, “Justesen’s construction--The low-rate case (Corresp.)”, IEEE Transactions on Information Theory 19, 711 (1973) DOI
A. Couvreur, H. Randriambololona, "Algebraic Geometry Codes and Some Applications." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
A. Vardy and Y. Be’ery, “Bit-level soft-decision decoding of Reed-Solomon codes”, IEEE Transactions on Communications 39, 440 (1991) DOI
S. A. Aly, A. Klappenecker, and P. K. Sarvepalli, “Subsystem Codes”, (2006) arXiv:quant-ph/0610153
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“Bose–Chaudhuri–Hocquenghem (BCH) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_q-ary_bch, title={Bose–Chaudhuri–Hocquenghem (BCH) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Bose–Chaudhuri–Hocquenghem (BCH) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.