[Jump to code hierarchy]

\(q\)-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 \(q\).

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.

Member of code lists

Primary Hierarchy

Classical Domain
Ring Kingdom
Ring codeECC
Parents
Ring code
\(q\)-ary code over \(\mathbb{Z}_q\)
Children
Binary codeGray Combinatorial design Nearly perfect Perfect binary
A \(q\)-ary code over \(\mathbb{Z}_q\) reduces to a binary code at \(q=2\). Ternary computing may be more applicable than binary computing to cryptographic schemes [2,3].
Error-correcting output code (ECOC)
\(q\)-ary additive code over \(\mathbb{Z}_q\)Linear code over \(\mathbb{Z}_q\) Gray

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
[2]
B. Cambou and D. Telesca, “Ternary Computing to Strengthen Cybersecurity”, Advances in Intelligent Systems and Computing 898 (2018) DOI
[3]
S. Assiri, B. Cambou, D. D. Booher, D. Ghanai Miandoab, and M. Mohammadinodoushan, “Key Exchange using Ternary system to Enhance Security”, 2019 IEEE 9th Annual Computing and Communication Workshop and Conference (CCWC) (2019) DOI
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

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} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/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

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