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
- \((5,40,4)_{\mathbb{Z}_4}\) 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 \(GF(q)\) 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 \(G=GF(q)\) since Galois fields are Abelian groups under addition.
Additive \(q\)-ary code
Children
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]
- 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 symplectic group and optimal EAQECC codes”, (2025) arXiv:2501.15465
Page edit log
- Victor V. Albert (2022-05-19) — most recent
- Shuubham Ojha (2022-05-18)
Cite as:
“Additive \(q\)-ary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_additive