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Register a new userSome results of linear codes over the ring $mathbb{Z}_4+umathbb{Z}_4+vmathbb{Z}_4+uvmathbb{Z}_4$
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Added by
Xuemei Guo
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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In this paper, we mainly study the theory of linear codes over the ring $R =mathbb{Z}_4+umathbb{Z}_4+vmathbb{Z}_4+uvmathbb{Z}_4$. By the Chinese Remainder Theorem, we have $R$ is isomorphic to the direct sum of four rings $mathbb{Z}_4$. We define a Gray map $Phi$ from $R^{n}$ to $mathbb{Z}_4^{4n}$, which is a distance preserving map. The Gray image of a cyclic code over $R^{n}$ is a linear code over $mathbb{Z}_4$. Furthermore, we study the MacWilliams identities of linear codes over $R$ and give the the generator polynomials of cyclic codes over $R$. Finally, we discuss some properties of MDS codes over $R$.
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Linear codes are considered over the ring $mathbb{Z}_4+vmathbb{Z}_4$, where $v^2=v$. Gray weight, Gray maps for linear codes are defined and MacWilliams identity for the Gray weight enumerator is given. Self-dual codes, construction of Euclidean isodual codes, unimodular complex lattices, MDS codes and MGDS codes over $mathbb{Z}_4+vmathbb{Z}_4$ are studied. Cyclic codes and quadratic residue codes are also considered. Finally, some examples for illustrating the main work are given.
We apply quantum Construction X on quasi-cyclic codes with large Hermitian hulls over $mathbb{F}_4$ and $mathbb{F}_9$ to derive good qubit and qutrit stabilizer codes, respectively. In several occasions we obtain quantum codes with stricly improved parameters than the current record. In numerous other occasions we obtain quantum codes with best-known performance. For the qutrit ones we supply a systematic construction to fill some gaps in the literature.
We consider the braid group representation which describes the non-abelian braiding statistics of the spin 1/2 particle world lines of an SU(2)$_4$ Chern-Simons theory. Up to an abelian phase, this is the same as the non-Abelian statistics of the elementary quasiparticles of the $k=4$ Read-Rezayi quantum Hall state. We show that these braiding statistics are identical to those of Z$_3$ Parafermions.
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring $R=F_{q}+vF_{q}+v^{2}F_{q}$, where $v^{3}=v$, for $q$ odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over $R$. Further, we give bounds on the minimum distance of LCD codes over $F_q$ and extend these to codes over $R$.
Quantum synchronizable codes are kinds of quantum error-correcting codes that can not only correct the effects of quantum noise on qubits but also the misalignment in block synchronization. This paper contributes to constructing two classes of quantum synchronizable codes by the cyclotomic classes of order two over $mathbb{Z}_{2q}$, whose synchronization capabilities can reach the upper bound under certain conditions. Moreover, the quantum synchronizable codes possess good error-correcting capability towards bit errors and phase errors.
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