# 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

## 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.