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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 \(q\).

Protection

In addition to the Hamming distance, codes over \(\mathbb{Z}_q\) are also defined over the Lee metric [1]. Code bounds exist under this metric [2].

Notes

See books [3,4] for introductions.

Cousins

  • Combinatorial design— Optimal constant-weight codes over \(\mathbb{Z}_q\) can be constructed [5] from a generalization of combinatorial designs to \(q\)-ary alphabets [6,7].
  • Modular-qudit code— Modular-qudit codes are quantum counterparts of \(q\)-ary codes over \(\mathbb{Z}_q\).
  • Julin-Golay code— Julin codes can be obtained from simple nonlinear codes over \(\mathbb{Z}_4\) using the Gray map [8].
  • Constant-weight block code— Optimal constant-weight codes over \(\mathbb{Z}_q\) can be constructed [5] from a generalization of combinatorial designs to \(q\)-ary alphabets [6,7].
  • \(q\)-ary code— \(q\)-ary codes for \(q=p\) prime are \(p\)-ary codes over \(\mathbb{Z}_p \cong \mathbb{F}_p\).
  • Polyphase code— Polyphase codes are spherical codes that can be obtained from \(q\)-ary codes over rings \(\mathbb{Z}_q\).

Member of code lists

Primary Hierarchy

Classical Domain
Ring Kingdom
Ring codeECC
Parents
Ring code
Homogeneous-space codeECC
The space of \(q\)-ary codes over \(\mathbb{Z}_q\) under the Lee metric can be viewed as a finite symmetric space \(G/H\) with \(G = D_q \wr S_n\) [2][9; Table 3].
\(q\)-ary code over \(\mathbb{Z}_q\)
Children
Binary codeGray Constant-weight 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 [10,11].
Error-correcting output code (ECOC)
\((5,40,4)_{\mathbb{Z}_4}\) Pentacode
Linear code over \(\mathbb{Z}_q\)Gray

References

[1]
C. Lee, “Some properties of nonbinary error-correcting codes”, IEEE Transactions on Information Theory 4, 77 (1958) DOI
[2]
J. ASTOLA, “THE LEE-SCHEME AND BOUNDS FOR LEE-CODES”, Cybernetics and Systems 13, 331 (1982) DOI
[3]
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
[4]
R. Roth, Introduction to Coding Theory (Cambridge University Press, 2006) DOI
[5]
T. Etzion, “Optimal constant weight codes over Zk and generalized designs”, Discrete Mathematics 169, 55 (1997) DOI
[6]
H. Hanani, “On Some Tactical Configurations”, Canadian Journal of Mathematics 15, 702 (1963) DOI
[7]
S.-T. Xia and F.-W. Fu, “Undetected error probability of q-ary constant weight codes”, Designs, Codes and Cryptography 48, 125 (2007) DOI
[8]
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
[9]
C. Bachoc, D. C. Gijswijt, A. Schrijver, and F. Vallentin, “Invariant Semidefinite Programs”, International Series in Operations Research & Management Science 219 (2011) arXiv:1007.2905 DOI
[10]
B. Cambou and D. Telesca, “Ternary Computing to Strengthen Cybersecurity”, Advances in Intelligent Systems and Computing 898 (2018) DOI
[11]
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.