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Published in: Designs, Codes and Cryptography 2/2022

19-01-2022

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

Authors: Soumak Biswas, Maheshanand Bhaintwal

Published in: Designs, Codes and Cryptography | Issue 2/2022

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Abstract

A new class of binary quantum codes from cyclic codes over \(\mathbb {Z}_2\mathbb {Z}_2[u]/\langle u^4 \rangle \), \(u^4=0\), is introduced. The generator polynomials of \(\mathbb {Z}_2\mathbb {Z}_2[u]/\langle u^4 \rangle \)-cyclic codes of length (rs) are obtained through the factorization of \(x^r-1\) and \(x^s-1\) into pairwise coprime monic polynomials over \(\mathbb {Z}_2\), where r and s are odd positive integers. A minimal spanning set for these codes is obtained. Under some restricted conditions, the structure of the duals of \(\mathbb {Z}_2\mathbb {Z}_2[u]/\langle u^4\rangle \)-cyclic codes is also determined. Necessary and sufficient conditions for a \(\mathbb {Z}_2\mathbb {Z}_2[u]/\langle u^4 \rangle \)-cyclic code of this restricted class to contain its dual or to be self-orthogonal are obtained. A new Gray map is defined, and the binary quantum codes are obtained by using the Calderbank-Shor-Steane construction on self-orthogonal or dual containing \(\mathbb {Z}_2\mathbb {Z}_2[u]/\langle u^4 \rangle \)-cyclic codes. Some examples of binary quantum codes with good parameters constructed from \(\mathbb {Z}_2\mathbb {Z}_2[u]/\langle u^4 \rangle \)-cyclic codes are given.
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Metadata
Title
Quantum codes from -cyclic codes
Authors
Soumak Biswas
Maheshanand Bhaintwal
Publication date
19-01-2022
Publisher
Springer US
Published in
Designs, Codes and Cryptography / Issue 2/2022
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-021-00978-1

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