# Hergert code[1]

Also known as Goethals-Delsarte (GD) code.

## Description

A nonlinear subcode of an RM code that is a formal dual of the nonlinear DG code in the sense that its distance distribution is equal to the MacWilliams transform of the distance distribution of a DG codes.

The Hergert code for \( m/2 \geq r \geq 2 \) is a nonlinear subcode of RM\((m-2,m)\), and equals RM\((m-2,m)\) at \(r=1\) [1; Thm. 2]. For each DG\((m,r)\) code, the Hergert code is defined as the union of cosets of RM\((m-3,m)\) in RM\((m-2,m)\), with coset representatives obtained by applying a particular linear bijection to the coset representatives of the DG code [1].

## Decoding

Since the equivalent \(\mathbb{Z_4}\) codes are extended cyclic codes, efficient encoding and decoding is possible. [2,3].

## Parent

## Children

- Goethals code — Goethals codes are equivalent to Hergert codes for \(r=3\) [1; Thm. 2].
- Preparata code — Preparata codes are equivalent to Hergert codes for \(r=2\) [1; Thm. 2].

## Cousins

- Delsarte-Goethals (DG) code — Hergert codes are duals of DG codes in that their distance distribution is equal to the MacWilliams transform of the distance distribution of DG codes [4]. However, the two codes are images of a pair of mutually dual linear codes over \(\mathbb{Z}_4\) under the Gray map [2,5].
- Dual linear code — Hergert codes are duals of DG codes in that their distance distribution is equal to the MacWilliams transform of the distance distribution of DG codes [4]. However, the two codes are images of a pair of mutually dual linear codes over \(\mathbb{Z}_4\) under the Gray map [2,5].
- Quaternary linear code over \(\mathbb{Z}_4\) — Hergert codes can be seen, via the Gray map, as linear codes over \(\mathbb{Z}_4\) [2,5].
- Gray code — Hergert codes can be seen, via the Gray map, as linear codes over \(\mathbb{Z}_4\) [2,5].

## References

- [1]
- P. Delsarte and J. M. Goethals, “Alternating bilinear forms over GF(q)”, Journal of Combinatorial Theory, Series A 19, 26 (1975) DOI
- [2]
- A. R. Hammons et al., “The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes”, IEEE Transactions on Information Theory 40, 301 (1994) DOI
- [3]
- T. Helleseth and P. V. Kumar, “The algebraic decoding of the Z/sub 4/-linear Goethals code”, IEEE Transactions on Information Theory 41, 2040 (1995) DOI
- [4]
- F. B. Hergert, “On the delsarte-goethals codes and their formal duals”, Discrete Mathematics 83, 249 (1990) DOI
- [5]
- A. R. Hammons Jr. et al., “The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes”, (2002) arXiv:math/0207208

## Page edit log

- Victor V. Albert (2024-01-11) — most recent

## Cite as:

“Hergert code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/hergert