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\(q\)-ary linear code over \(\mathbb{Z}_q\)

Description

A code encoding \(K\) states (codewords) in \(n\) coordinates over the ring \(\mathbb{Z}_q\) of integers modulo \(q\) that is closed under codeword addition and scalar multiplication.

Cousins

  • Constantin-Rao (CR) code— CR codes, and their special cases the VT codes, can be converted to ternary codes with nice structure via a binary-to-ternary map \(00\to 0\), \(11\to 0\), \(01\to 1\), and \(10\to 2\) [1].
  • Bose–Chaudhuri–Hocquenghem (BCH) code— BCH codes for \(q=p\) prime field can also work as codes over the Lee metric [2].
  • Modular-qudit CSS code— The modular-qudit CSS construction uses two related \(q\)-ary linear codes over \(\mathbb{Z}_q\), \(C_X\) and \(C_Z\).

Member of code lists

Primary Hierarchy

Classical Domain
Ring Kingdom
Ring codeECC
\(q\)-ary code over \(\mathbb{Z}_q\)
\(q\)-ary additive code over \(\mathbb{Z}_q\)
Parents
\(q\)-ary additive code over \(\mathbb{Z}_q\)
For \(q>2\), additive codes need not be linear since linearity also requires closure under multiplication.
\(R\)-linear codeECC
\(q\)-ary linear code over \(\mathbb{Z}_q\)
Children
Linear binary codeGray
Linear binary codes are linear \(q\)-ary codes over \(\mathbb{Z}_q\) for \(q=2\).
Pless symmetry code
Ternary Golay code
Tetracode
\([56,6,36]_3\) Hill-cap code
Classical fractal liquid code
Berlekamp code
Quaternary linear code over \(\mathbb{Z}_4\)

References

[1]
M. Grassl, P. W. Shor, G. Smith, J. Smolin, and B. Zeng, “New Constructions of Codes for Asymmetric Channels via Concatenation”, IEEE Transactions on Information Theory 61, 1879 (2015) arXiv:1310.7536 DOI
[2]
R. M. Roth and P. H. Siegel, “Lee-metric BCH codes and their application to constrained and partial-response channels”, IEEE Transactions on Information Theory 40, 1083 (1994) DOI
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Zoo Code ID: q-ary_linear_over_zq

Cite as:
“\(q\)-ary linear code over \(\mathbb{Z}_q\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_linear_over_zq
BibTeX:
@incollection{eczoo_q-ary_linear_over_zq, title={\(q\)-ary linear 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_linear_over_zq} }
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Permanent link:
https://errorcorrectionzoo.org/c/q-ary_linear_over_zq

Cite as:

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

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