Bose–Chaudhuri–Hocquenghem (BCH) code[1]
Description
Cyclic \(q\)-ary code, with \(n\) and \(q\) relatively prime, 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 \(r\geq 2\). More general BCH codes can be defined for zeroes are powers of the form \(\{b,b+s,b+2s,\cdots,b+(\delta-2)s\}\) where gcd\((s,n)=1\).
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
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.
Rate
Primitive BCH codes are asymptotically bad [2; pg. 269].
Decoding
Berlekamp-Massey decoder with runtime of order \(O(n^2)\) [3–5] 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].
Realizations
DVDs, disk drives, and two-dimensional bar codes [13].
Notes
See books [2,14,15] for expositions on BCH codes and code tables.See Kaiserslautern database [16] for explicit codes.See corresponding MinT database entry [17].
Parents
- Cyclic linear \(q\)-ary code
- Twisted BCH code — Twisted BCH codes are obtained from \(q\)-ary BCH codes via various lengthening and extension procedures.
Child
Cousins
- Reed-Solomon (RS) code — BCH codes are subfield subcodes of RS codes.
- Classical Goppa code — Narrow-sense BCH codes are Goppa codes with \(L=\{1,\alpha^{-1},\cdots,\alpha^{1-n}\}\) and \(G(x)=x^{\delta-1}\) ([15], pg. 522).
- \(q\)-ary linear LTC — Duals of BCH codes are locally testable [18].
- Justesen code — Using more general BCH codes instead of RS codes can improve the parameters of the Justesen codes [19].
- Reed-Solomon (RS) code — Narrow-sense RS codes are BCH codes [20; Remark 15.3.21][15; Thms. 5.2.1 and 5.2.3]. Their minimal distance is equal to their designed distance [21; 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 [22].
- Qubit BCH code — BCH codes are used to construct qubit BCH codes via the CSS construction or the Hermitian construction.
- Subsystem qubit stabilizer code — BCH codes yield subsystem stabilizer codes via the subsystem extension of the Hermitian construction to subsystem codes [23; Exam. 1].
- Galois-qudit BCH code
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]
- S. Zhu, Z. Sun, and X. Kai, “A Class of Narrow-Sense BCH Codes”, IEEE Transactions on Information Theory 65, 4699 (2019) DOI
- [14]
- S. Lin and D. J. Costello, Error Control Coding, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 2004.
- [15]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [16]
- 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.
- [17]
- 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
- [18]
- T. Kaufman and S. Litsyn, “Almost Orthogonal Linear Codes are Locally Testable”, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) DOI
- [19]
- E. J. Weldon, “Justesen’s construction--The low-rate case (Corresp.)”, IEEE Transactions on Information Theory 19, 711 (1973) DOI
- [20]
- A. Couvreur, H. Randriambololona, "Algebraic Geometry Codes and Some Applications." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [21]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [22]
- A. Vardy and Y. Be’ery, “Bit-level soft-decision decoding of Reed-Solomon codes”, IEEE Transactions on Communications 39, 440 (1991) DOI
- [23]
- S. A. Aly, A. Klappenecker, and P. K. Sarvepalli, “Subsystem Codes”, (2006) arXiv:quant-ph/0610153
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