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.

Child

  • Linear \(q\)-ary code — For \(q>2\), additive codes need not be linear since linearity also requires closure under multiplication.

Cousins

  • Stabilizer code over \(GF(4)\) — Let \(\phi\) be a bijection from a linear binary subspace to \(GF(4)^n\). Let \(C\) be an additive self-orthogonal subcode over \(GF(4)\), containing \(2^{n-k}\) vectors, such that there are no vectors of weight less than \(d\) in \(C^{\perp}\setminus C\). Then, any eigenspace of the inverse map \(\phi^{-1}(C)\) is an \([[n, k, d]]\) stabilizer code over \(GF(4)\).
  • Galois-qudit stabilizer code — A Galois-qudit stabilizer code is the closest quantum analogue of an additive code over \(GF(q)\) because addition in the field corresponds to multiplication of stabilizers in the quantum case.
  • Stabilizer code over \(GF(q^2)\) — Self-orthogonal additive \(q\)-ary codes are used in this construction.

Zoo code information

Internal code ID: q-ary_additive

Your contribution is welcome!

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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} }
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/tree/main/codes/classical/q-ary_digits/q-ary_additive.yml.