Description
A \(q\)-ary code whose codewords are closed under addition, i.e., for any codewords \(x,y\), \(x+y\) is also a codeword. If \(q=p^m\), then additive closure already implies \(\mathbb{F}_p\)-linearity because multiplying a codeword by any \(\lambda\in\mathbb{F}_p\) is equivalent to adding that codeword to itself \(\lambda\) times.Cousins
- \((5,40,4)_{\mathbb{Z}_4}\) Pentacode— A close relative of the pentacode is an additive quaternary code [1].
- EA qubit stabilizer code— There is a relation between quaternary additive codes and EA qubit stabilizer codes [2].
- Galois-qudit stabilizer code— Galois-qudit stabilizer codes are the closest quantum analogues of additive codes over \(\mathbb{F}_q\) because addition in the field corresponds to multiplication of stabilizers in the quantum case.
Member of code lists
Primary Hierarchy
Parents
Additive \(q\)-ary codes are linear over the additive group \(G=\mathbb{F}_q\). If \(q=p^m\), they are always \(\mathbb{F}_p\)-linear, but for \(m>1\) they need not be \(\mathbb{F}_q\)-linear.
Additive \(q\)-ary code
Children
Linear \(q\)-ary codeGray Evaluation MDS GRS Self-dual linear GRM QR Projective geometry Quantum-inspired classical block Tanner \(q\)-ary LDPC Divisible
For \(q>2\), additive codes need not be linear since linearity also requires closure under multiplication.
References
- [1]
- M. Ran and J. Snyders, “On cyclic reversible self-dual additive codes with odd length over Z/sub 2//sup 2/”, IEEE Transactions on Information Theory 46, 1056 (2000) DOI
- [2]
- R. Li, Y. Ren, C. Guan, and Y. Liu, “Geometry of the symplectic group and optimal EAQECC codes”, (2025) arXiv:2501.15465
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2022-05-19)
- Shuubham Ojha (2022-05-18)
Cite as:
“Additive \(q\)-ary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/q-ary_additive, arXiv:2606.11484