New Codes from Cyclic and Negacyclic Codes of Even Length over $\mathbb{Z}_4$ New Codes from Cyclic and Negacyclic Codes of Even Length over $\mathbb{Z}_4$

Nuh Aydın, Mohamed O. Belghith, Godwin Idowu +2 more

This paper uses theoretical results previously established in the literature to design search algorithms to find new linear codes over $\mathbb{Z}_4$ from cyclic and negacyclic codes of even length. As a result of these searches, we have found 2500 new cyclic codes and 730 negacyclic codes. These new codes exhibit improved parameters compared to previously known codes. Additionally, we have obtained binary quantum codes with good parameters from such $\mathbb{Z}_4$ codes.This paper uses theoretical results previously established in the literature to design search algorithms to find new linear codes over $\mathbb{Z}_4$ from cyclic and negacyclic codes of even length. As a result of these searches, we have found 2500 new cyclic codes and 730 negacyclic codes. These new codes exhibit improved parameters compared to previously known codes. Additionally, we have obtained binary quantum codes with good parameters from such $\mathbb{Z}_4$ codes.

arXiv:2606.08750Jun 7, 20260 citations

New quantum codes from self-dual codes over $\mathbb {F}_4$ New quantum codes from self-dual codes over $$\mathbb {F}_4$$

Reza Dastbasteh, Petr Lisoněk

Designs Codes and CryptographyOct 13, 202312 citations

Partial Permutation Decoding for Several Families of Linear and ${\mathbb{Z}}_4$ -Linear CodesPartial Permutation Decoding for Several Families of Linear and ${\mathbb{Z}}_4$ -Linear Codes

Roland D. Barrolleta, Mercè Villanueva

A general criterion to obtain s-PD-sets of minimum size s + 1 for partial permutation decoding, which enable correction up to s errors, for systematic codes over a finite field Fq and Z4-linear codes is provided. We show how this technique can be easily applied to linear cyclic codes over Fq, Z4-linear codes which are the Gray map image of a quaternary linear cyclic code, and some related codes such as quasi-cyclic codes. Furthermore, specific results for some linear and nonlinear binary codes, including simplex, Kerdock, Delsarte-Goethals, and extended dualized Kerdock codes are given. Finally, applying this technique, new s-PD-sets of size s + 1 for Z4-linear Hadamard codes of type 2γ4δ, for all δ ≥ 4 and 1δ- 3; and for Z4-linear simplex codes of type 4m, for all m ≥ 2 and 1m+1- 3, are also provided.A general criterion to obtain s-PD-sets of minimum size s + 1 for partial permutation decoding, which enable correction up to s errors, for systematic codes over a finite field Fq and Z4-linear codes is provided. We show how this technique can be easily applied to linear cyclic codes over Fq, Z4-linear codes which are the Gray map image of a quaternary linear cyclic code, and some related codes such as quasi-cyclic codes. Furthermore, specific results for some linear and nonlinear binary codes, including simplex, Kerdock, Delsarte-Goethals, and extended dualized Kerdock codes are given. Finally, applying this technique, new s-PD-sets of size s + 1 for Z4-linear Hadamard codes of type 2γ4δ, for all δ ≥ 4 and 1δ- 3; and for Z4-linear simplex codes of type 4m, for all m ≥ 2 and 1m+1- 3, are also provided.

IEEE Transactions on Information TheoryMay 24, 20188 citations

Quantum codes from $\mathbb {Z}_2\mathbb {Z}_2[u]/\langle u^4 \rangle$ -cyclic codesQuantum codes from $$\mathbb {Z}_2\mathbb {Z}_2[u]/\langle u^4 \rangle $$-cyclic codes

Soumak Biswas, Maheshanand Bhaintwal

Designs Codes and CryptographyJan 18, 20227 citations

SISO Decoding of ℤ4 Linear Kerdock and Preparata Codes

Aleksandar Minja, V. Šenk

Some nonlinear codes, such as Kerdock and Preparata codes, can be represented as binary images under the Gray map of linear codes over rings. This paper introduces MAP decoding of Kerdock and Preparata codes by working with their quaternary representation (linear codes over $\mathbb {Z}_{4}$ ) with the complexity of $\mathcal {O}(N^{2}\log _{2} N)$ , where N is the code length in $\mathbb {Z}_{4}$ . A sub-optimal bitwise APP decoder with good error-correcting performance and complexity of $\mathcal {O}(N\log _{2} N)$ that is constructed using the decoder lifting technique is also introduced. This APP decoder extends upon the original lifting decoder by working with likelihoods instead of hard decisions and is not limited to Kerdock and Preparata code families. Simulations show that our novel decoders significantly outperform several popular decoders in terms of error rate.Some nonlinear codes, such as Kerdock and Preparata codes, can be represented as binary images under the Gray map of linear codes over rings. This paper introduces MAP decoding of Kerdock and Preparata codes by working with their quaternary representation (linear codes over$\mathbb {Z}_{4}$) with the complexity of$\mathcal {O}(N^{2}\log _{2} N)$, where N is the code length in$\mathbb {Z}_{4}$. A sub-optimal bitwise APP decoder with good error-correcting performance and complexity of$\mathcal {O}(N\log _{2} N)$that is constructed using the decoder lifting technique is also introduced. This APP decoder extends upon the original lifting decoder by working with likelihoods instead of hard decisions and is not limited to Kerdock and Preparata code families. Simulations show that our novel decoders significantly outperform several popular decoders in terms of error rate.

arXiv:2312.00193IEEE Transactions on CommunicationsJan 11, 20225 citations

Quantum Codes Over an Extension of

{\mathbb {Z}_4}

Quantum Cyclic Codes Over ℤ m

{\mathbb{Z}}_m

New Codes from Cyclic and Negacyclic Codes of Even Length over

\mathbb{Z}_4

This paper uses theoretical results previously established in the literature to design search algorithms to find new linear codes over

\mathbb{Z}_4

from cyclic and negacyclic codes of even length. As a result of these searches, we have found 2500 new cyclic codes and 730 negacyclic codes. These new codes exhibit improved parameters compared to previously known codes. Additionally, we have obtained binary quantum codes with good parameters from such

\mathbb{Z}_4

codes.

New quantum codes from self-dual codes over

\mathbb {F}_4

Partial Permutation Decoding for Several Families of Linear and

{\mathbb{Z}}_4

-Linear Codes

Quantum codes from

\mathbb {Z}_2\mathbb {Z}_2[u]/\langle u^4 \rangle

-cyclic codes

\mathbb {Z}_{4}

) with the complexity of

\mathcal {O}(N^{2}\log _{2} N)

, where N is the code length in

\mathbb {Z}_{4}

. A sub-optimal bitwise APP decoder with good error-correcting performance and complexity of

\mathcal {O}(N\log _{2} N)

that is constructed using the decoder lifting technique is also introduced. This APP decoder extends upon the original lifting decoder by working with likelihoods instead of hard decisions and is not limited to Kerdock and Preparata code families. Simulations show that our novel decoders significantly outperform several popular decoders in terms of error rate.