## Description

An additive \((n,2^n)_q\) code that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to some inner product, usually the trace-Hermitian inner product.

Self-dual additive codes that contain at least one codeword of odd weight are called Type I additive. Even self-dual additive codes are called Type II additive, existing only for even \(n\) [1; Sec. 9.10]. Type I (type II) additive codes with length up to seven (eight) have been classified [2]. Much is known about codes up to length 16 [3].

## Protection

The minimum distance of a trace-Hermitian self-dual additive \((n,n/2\) code satisfies [4; Thm. 33] \begin{align} d\leq\begin{cases} 2\left\lfloor \frac{n}{6}\right\rfloor +1 & n\equiv0\text{ mod 6 and code is Type I additive}\\ 4\left\lfloor \frac{n}{6}\right\rfloor +3 & n\equiv5\text{ mod 6 and code is Type I additive}\\ 2\left\lfloor \frac{n}{6}\right\rfloor +2 & \text{otherwise for Type I additive codes}\\ 2\left\lfloor \frac{n}{6}\right\rfloor +2 & \text{code is Type II additive} \end{cases}~. \tag*{(1)}\end{align} A self-dual additive code saturating the above inequality is called extremal additive.

## Parent

## Children

- Self-dual linear code — Self-dual linear codes with respect to some inner product are automatically self-dual additive under the same inner product since linear codes are additive. In addition, quaternary linear codes are Hermitian self-orthogonal (self-dual) iff they are trace-Hermitian self-orthogonal (self-dual) additive [5; Thm. 27.4.1] ([1; Thm. 9.10.3]).
- Dodecacode — The dodecacode is trace-Hermitian self-dual additive.

## Cousins

- Dual linear code — The difference between the definitions of dual linear and dual additive codes is in the trace used in the inner product.
- Unimodular lattice code — There are parallels between self-dual additive codes over \(\mathbb{Z}_{2k}\) and even unimodular lattices [6].

## References

- [1]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [2]
- G. Hoehn, “Self-dual Codes over the Kleinian Four Group”, (2000) arXiv:math/0005266
- [3]
- P. Gaborit, W. C. Huffman, J.-L. Kim, and V. Pless, “On additive GF(4) codes,” in Codes and Association Schemes (DIMACS Workshop, November 9–12, 1999), eds. A. Barg and S. Litsyn. Providence, RI: American Mathematical Society, 2001, pp. 135–149.
- [4]
- E. M. Rains and N. J. A. Sloane, “Self-dual codes,” in Handbook of Coding Theory, eds. V. S. Pless and W. C. Huffman. Amsterdam: Elsevier, 1998, pp. 177–294.
- [5]
- M. F. Ezerman, "Quantum Error-Control Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [6]
- E. Bannai et al., “Type II codes, even unimodular lattices, and invariant rings”, IEEE Transactions on Information Theory 45, 1194 (1999) DOI

## Page edit log

- Victor V. Albert (2022-07-22) — most recent

## Cite as:

“Self-dual additive code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/self_dual_additive