Primitive narrow-sense BCH code[1]
Description
BCH codes for \(b=1\) and for \(n=q^r-1\) for some \(r\geq 2\).
Parents
- Bose–Chaudhuri–Hocquenghem (BCH) code — BCH codes are called narrow-sense when \(b=1\), and are called primitive when \(n=q^r-1\) for some \(r\geq 2\).
- Goppa code — Primitive narrow-sense BCH codes are Goppa codes with \(L=\{1,\alpha^{-1},\cdots,\alpha^{1-n}\}\) and \(G(x)=x^{\delta-1}\) [2; pg. 522].
Child
- \([2^r-1,2^r-r-1,3]\) Hamming code — Binary Hamming codes are binary primitive narrow-sense BCH codes [2; Corr. 5.1.5]. Binary Hamming codes can be written in cyclic form [3; Thm. 12.22].
Cousin
- Quantum data-syndrome (QDS) code — Primitive narrow-sense BCH codes can be used as the syndrome measurement codes of a QDS code [4]. This construction requires fewer measurements than a previous general construction [5].
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]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [3]
- R. Hill. A First Course In Coding Theory. Oxford University Press, 1988.
- [4]
- E. Guttentag, A. Nemec, and K. R. Brown, “Robust Syndrome Extraction via BCH Encoding”, (2023) arXiv:2311.16044
- [5]
- Y. Fujiwara, “Ability of stabilizer quantum error correction to protect itself from its own imperfection”, Physical Review A 90, (2014) arXiv:1409.2559 DOI
Page edit log
- Victor V. Albert (2024-08-13) — most recent
Cite as:
“Primitive narrow-sense BCH code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/narrow_sense_q-ary_bch