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Additive \(q\)-ary code

Description

A \(q\)-ary code whose codewords are closed under addition, i.e., for any codewords \(x,y\), \(x+y\) is also a codeword.

Cousins

  • EA qubit stabilizer code— There is a relation between quaternary additive codes and EA qubit stabilizer codes [1].
  • Galois-qudit stabilizer code— Galois-qudit stabilizer codes are the closest quantum analogues of additive codes over \(GF(q)\) because addition in the field corresponds to multiplication of stabilizers in the quantum case.

Member of code lists

Primary Hierarchy

Classical Domain
Galois-field Kingdom
\(q\)-ary codeECC
Parents
\(q\)-ary code
Linear code over \(G\)ECC
Additive \(q\)-ary codes are linear over \(G=GF(q)\) since Galois fields are Abelian groups under addition.
Additive \(q\)-ary code
Children
Twisted BCH code
Dual additive codeSelf-dual additive Self-dual linear
Linear \(q\)-ary codeGray Evaluation MDS GRS Self-dual linear GRM QR Projective geometry Tanner \(q\)-ary LDPC Divisible
For \(q>2\), additive codes need not be linear since linearity also requires closure under multiplication.

References

[1]
R. Li, Y. Ren, C. Guan, and Y. Liu, “Geometry of symplectic group and optimal EAQECC codes”, (2025) arXiv:2501.15465
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Zoo Code ID: q-ary_additive

Cite as:
“Additive \(q\)-ary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_additive
BibTeX:
@incollection{eczoo_q-ary_additive, title={Additive \(q\)-ary code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/q-ary_additive} }
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Permanent link:
https://errorcorrectionzoo.org/c/q-ary_additive

Cite as:

“Additive \(q\)-ary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_additive

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