\(q\)-ary code over \(\mathbb{Z}_q\) 

Description

A code encoding \(K\) states (codewords) in \(n\) coordinates over the ring \(\mathbb{Z}_q\) of integers modulo \(q\).

Parent

Child

Cousins

  • Julin-Golay code — Julin codes can be obtained from simple nonlinear codes over \(\mathbb{Z}_4\) using the Gray map [1].
  • Polyphase code — Polyphase codes are spherical codes that can be obtained from \(q\)-ary codes over rings \(\mathbb{Z}_q\).
  • Modular-qudit stabilizer code — Modular-qudit stabilizer codes are the closest quantum analogues of additive codes over \(\mathbb{Z}_q\) because addition in the ring corresponds to multiplication of stabilizers in the quantum case.

References

[1]
J. H. Conway and N. J. A. Sloane, “Quaternary constructions for the binary single-error-correcting codes of Julin, Best and others”, Designs, Codes and Cryptography 4, 31 (1994) DOI
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Zoo Code ID: q-ary_over_zq

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

“\(q\)-ary code over \(\mathbb{Z}_q\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_over_zq

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/rings/over_zq/q-ary_over_zq.yml.