Self-dual additive code 

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.

Parents

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
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Zoo Code ID: self_dual_additive

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/dual/self_dual_additive.yml.