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

Notes

See books [1,2] for introductions.

Cousins

  • Julin-Golay code— Julin codes can be obtained from simple nonlinear codes over \(\mathbb{Z}_4\) using the Gray map [3].
  • 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 [4,5].
Error-correcting output code (ECOC)
Linear code over \(\mathbb{Z}_q\)Gray

References

[1]
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
[2]
R. Roth, Introduction to Coding Theory (Cambridge University Press, 2006) DOI
[3]
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
[4]
B. Cambou and D. Telesca, “Ternary Computing to Strengthen Cybersecurity”, Advances in Intelligent Systems and Computing 898 (2018) DOI
[5]
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
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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.