Self-dual additive code 


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].


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.



  • 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.


  • 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].


W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
G. Hoehn, “Self-dual Codes over the Kleinian Four Group”, (2000) arXiv:math/0005266
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.
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.
M. F. Ezerman, "Quantum Error-Control Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
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

Your contribution is welcome!

on (edit & pull request)

edit on this site

Zoo Code ID: self_dual_additive

Cite as:
“Self-dual additive code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_self_dual_additive, title={Self-dual additive code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:

Cite as:

“Self-dual additive code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.