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Linear code over \(\mathbb{Z}_4\)

Description

A code that forms a subgroup of \(\mathbb{Z}_4^n\) under addition. More technically, linear codes over \(\mathbb{Z}_4\) are submodules of \(\mathbb{Z}_4^n\).

Linear codes over \(\mathbb{Z}_4\) can be defined in terms of a generator matrix \(G\), whose rows generate the \(k\)-dimensional codespace. Given a message (i.e., logical string) \(x\), the corresponding encoded codeword is \(G^T x\).

A code's generator matrix can be reduced via coordinate permutations to its standard form \begin{align} G=\left[\begin{array}{ccc} I_{k_{1}} & A & B\\ 0 & 2I_{k_{2}} & 2C \end{array}\right]~, \tag*{(1)}\end{align} where \(A,C\) are binary matrices, and where \(B\) is a quaternary matrix. Such a code encodes \(4^{k_1} 2^{k_2}\) codewords, consisting of \(k_1\) logical quaternary digits and \(k_2\) logical bits. It is called an \((n,4^{k_1} 2^{k_2})_{\mathbb{Z}_4}\) linear code over \(\mathbb{Z}_4\) of type \(4^{k_1} 2^{k_2}\). The generator matrix forms a basis when there are no logical bits, i.e., the submodule is free if and only if \(k_2=0\) [1].

Two linear codes over \(\mathbb{Z}_4\) are monomially equivalent if the codewords of one code can be mapped into those of the other under a permutation and coordinate sign flips. The monomial automorphism group of a linear code over \(\mathbb{Z}_4\) is the largest subgroup of coordinate permutations and sign flips that maps the code onto itself.

Protection

Quaternary codes over \(\mathbb{Z}_4\) come with a Hamming distance, which is the minimum number of nonzero coordinates of a nonzero codeword. They also protect from cyclic shifts in each quaternary digit, which are quantified by the Lee metric. The Lee weight of a digit is given by \(w_L(0)=0\), \(w_L(1)=1\), \(w_L(2)=2\), and \(w_L(3)=1\), and the Lee weight of a string is the integral sum of the Lee weights of its digits. The code's Lee distance of a code is the minimum Lee weight of a nonzero codeword.

The Lee distance \(d_L\) between two strings \(a,b\in\mathbb{Z}_4\) governs how many cyclic shifts it takes to connect the two strings, and is thus related to the Euclidean distance between the vectors \(i^a\) and \(i^b\) as \(2d_L(a,b) = \|i^a - i^b\|^2\). Quaternary codes over the Lee metric are thus naturally mapped to QPSK codes.

Weight enumerators have also been extended to the Lee metric [2].

Notes

Code Database, including quasi-cyclic and quasi-twisted codes [3].See books [1,4,5] for introductions.

Cousins

  • Quadrature PSK (QPSK) code— The Lee distance \(d_L\) between two strings \(a,b\in\mathbb{Z}_4\) governs how many cyclic shifts it takes to connect the two strings, and is thus related to the Euclidean distance between \(i^a\) and \(i^b\) as \(2d_L(a,b) = \|i^a - i^b\|^2\). Quaternary codes over the Lee metric are thus naturally mapped to QPSK codes.
  • Construction \(A_4\) code— Every linear code over \(\mathbb{Z}_4\) yields a lattice under Construction \(A_4\) [4; Sec. 12.5.3].
  • Melas code— The even-weight subcode of the Melas code can be lifted to a linear code over \(\mathbb{Z}_4\) [6].
  • Hergert code— Hergert codes can be seen, via the Gray map, as linear codes over \(\mathbb{Z}_4\) [1,7,8].
  • Gray code— A linear code \(C\) over \(\mathbb{Z}_4\) can be mapped, via the Gray map, to a binary code. The binary code is linear if and only if doubling the component-wise product of any two codewords in \(C\) yields another codeword in \(C\) [4; Thm. 12.2.3].
  • Delsarte-Goethals (DG) code— DG codes can be seen, via the Gray map, as extended linear cyclic codes over \(\mathbb{Z}_4\) [7,8].

Member of code lists

Primary Hierarchy

Parents
Linear binary codes are linear \(q\)-ary codes over \(\mathbb{Z}_q\) for \(q=4\).
Linear code over \(\mathbb{Z}_4\)
Children
The extended quaternary Golay code is an extremal Type II self-dual code over \(\mathbb{Z}_4\) by virtue of its parameters [9].

References

[1]
Z. X. Wan, Quaternary Codes (WORLD SCIENTIFIC, 1997) DOI
[2]
M. Klemm, “Selbstduale Codes �ber dem Ring der ganzen Zahlen modulo 4”, Archiv der Mathematik 53, 201 (1989) DOI
[3]
N. Aydin, Y. Lu, and V. R. Onta, “An Updated Database of \(\mathbb{Z}_4\) Codes”, (2022) arXiv:2208.06832
[4]
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
[5]
R. Roth, Introduction to Coding Theory (Cambridge University Press, 2006) DOI
[6]
A. Alahmadi, H. Alhazmi, T. Helleseth, R. Hijazi, N. Muthana, and P. Solé, “On the lifted Melas code”, Cryptography and Communications 8, 7 (2015) DOI
[7]
A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole, “The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes”, IEEE Transactions on Information Theory 40, 301 (1994) DOI
[8]
A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, “The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes”, (2002) arXiv:math/0207208
[9]
A. Munemasa and R. A. L. Betty, “Classification of extremal type II \(\)\mathbb {Z}_4\(\)-codes of length 24”, Designs, Codes and Cryptography 92, 771 (2023) DOI
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Zoo Code ID: quaternary_over_z4

Cite as:
“Linear code over \(\mathbb{Z}_4\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quaternary_over_z4
BibTeX:
@incollection{eczoo_quaternary_over_z4, title={Linear code over \(\mathbb{Z}_4\)}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quaternary_over_z4} }
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“Linear code over \(\mathbb{Z}_4\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quaternary_over_z4

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/rings/over_zq/over_z4/linear_over_z4/quaternary_over_z4.yml.