Zetterberg code[1]

Description

Family of binary cyclic \([2^{2s}+1,2^{2s}-4s+1]\) codes with distance \(d>5\) generated by the minimal polynomial \(g_s(x)\) of \(\alpha\) over \(GF(2)\), where \(\alpha\) is a primitive \(n\)th root of unity in the field \(GF(2^{4s})\). They are quasi-perfect codes and are one of the best known families of double-error correcting binary linear codes

Protection

Correct at least all weight-2 errors.

Rate

The rate is given by \(1-\frac{4s}{n}\), which is asymptotically good, with a minimum distance of 5.

Decoding

Kallquist first described an algebraic decoding theorem [2]. A faster version was later provided in Ref. [3] and further modified in Ref. [4].

Realizations

Code used to provide better protection of data transmission with its double error correcting capacity [5].

Parents

Cousin

  • Perfect code — Zetterberg codes are quasi-perfect, with each \(n\)-bit string at most three bit-flips away from a codeword [3].

Zoo code information

Internal code ID: zetterberg

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: zetterberg

Cite as:
“Zetterberg code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/zetterberg
BibTeX:
@incollection{eczoo_zetterberg, title={Zetterberg code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/zetterberg} }
Permanent link:
https://errorcorrectionzoo.org/c/zetterberg

References

[1]
L.-H. Zetterberg, “Cyclic codes from irreducible polynomials for correction of multiple errors”, IEEE Transactions on Information Theory 8, 13 (1962). DOI
[2]
P. Kallquist, "Decoding of Zetterberg codes," in Proc. Fourth Joint Swedish-Soviet Workshop on Inform. Theory, Gotland, Sweden, Aug. 27-Sept. 1, 1989, p. 305-300
[3]
S. M. Dodunekov and J. E. M. Nilsson, “Algebraic decoding of the Zetterberg codes”, IEEE Transactions on Information Theory 38, 1570 (1992). DOI
[4]
M.-H. Jing et al., “A Result on Zetterberg Codes”, IEEE Communications Letters 14, 662 (2010). DOI
[5]
J. Meggitt, “Error correcting codes and their implementation for data transmission systems”, IEEE Transactions on Information Theory 7, 234 (1961). DOI

Cite as:

“Zetterberg code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/zetterberg

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/bits/zetterberg.yml.