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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 forms an Abelian subgroup of \(\mathbb{Z}_q^n\) under addition. Since addition of \(m\) identical elements is equivalent to multiplying by \(m\), linear codes over \(\mathbb{Z}_q\) are automatically closed under scalar multiplication. More technically, linear codes over \(\mathbb{Z}_q\) are submodules of \(\mathbb{Z}_q^n\).

Protection

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

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 \(10\to 2\) [2].
  • Bose–Chaudhuri–Hocquenghem (BCH) code— BCH codes for \(q=p\) prime field can also work as codes over the Lee metric [3].
  • 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 \(\mathbb{Z}_q\)
Parents
\(q\)-ary code over \(\mathbb{Z}_q\)
\(R\)-linear codeECC
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
Linear code over \(\mathbb{Z}_4\)
Linear binary codes are linear \(q\)-ary codes over \(\mathbb{Z}_q\) for \(q=4\).

References

[1]
C. Lee, “Some properties of nonbinary error-correcting codes”, IEEE Transactions on Information Theory 4, 77 (1958) DOI
[2]
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
[3]
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:
“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={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:

“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.