article · SIAM Journal on Discrete Mathematics · 2021

BCH Codes with Minimum Distance Proportional to Code Length

Satoshi Noguchi, Xiao‐Nan Lu, Masakazu Jimbo, Ying Miao

Cited by 21Abstract and details only

Abstract

BCH codes are among the best practical cyclic codes widely used in consumer electronics, communication systems, and storage devices. However, not much is known about BCH codes with large minimum distance. In this paper, we consider narrow-sense BCH codes of length $n = \frac{q^m-1}{N}$ with designed distance $\delta = \frac{s}{q-1}n$ proportional to $n$, where $N$ divides $\frac{q^m-1}{q-1}$ and $1 \le s \le q-1$. We determine both their dimensions and minimum distances. In particular, when $N=1$, the codes are primitive, with minimum distance $d=\frac{s}{q-1}(q^m-1)$ and dimension $k = (q-s)^m$. The general result on code dimensions is achieved by applying generating functions and inverse discrete Fourier transforms to an enumeration problem.

Cite this paper

APA 7

Noguchi, S., Lu, X., Jimbo, M., & Miao, Y. (2021). BCH Codes with Minimum Distance Proportional to Code Length. SIAM Journal on Discrete Mathematics, 35(1), 179–193. https://doi.org/10.1137/19m1260876

MLA 9

Noguchi, Satoshi, et al. “BCH Codes with Minimum Distance Proportional to Code Length.” SIAM Journal on Discrete Mathematics, vol. 35, no. 1, 2021, pp. 179–93, https://doi.org/10.1137/19m1260876.

Chicago

Noguchi, Satoshi, Xiao‐Nan Lu, Masakazu Jimbo, and Ying Miao. 2021. “BCH Codes with Minimum Distance Proportional to Code Length.” SIAM Journal on Discrete Mathematics 35 (1): 179–93. https://doi.org/10.1137/19m1260876.

Harvard

Noguchi, S. et al. (2021) ‘BCH Codes with Minimum Distance Proportional to Code Length’, SIAM Journal on Discrete Mathematics, 35(1), pp. 179–193. Available at: https://doi.org/10.1137/19m1260876.

BibTeX

@article{noguchi2021codes,
  title = {{BCH Codes with Minimum Distance Proportional to Code Length}},
  author = {Noguchi, Satoshi and Lu, Xiao‐Nan and Jimbo, Masakazu and Miao, Ying},
  journal = {SIAM Journal on Discrete Mathematics},
  volume = {35},
  number = {1},
  pages = {179--193},
  year = {2021},
  doi = {10.1137/19m1260876}
}

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