A note on cyclic codes over ℤ4 + uℤ4

2016 ◽  
Vol 08 (01) ◽  
pp. 1650017 ◽  
Author(s):  
Rama Krishna Bandi ◽  
Maheshanand Bhaintwal
Keyword(s):  
Local Ring ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Ideal Structure ◽  
Complete Ideal ◽  
Spanning Set

In this paper, we have studied cyclic codes over the ring [Formula: see text], [Formula: see text]. We have provided the general form of the generators of a cyclic code over [Formula: see text] and obtained a minimal spanning set for such codes and determined their ranks. We have determined a necessary condition and a sufficient condition for cyclic codes over [Formula: see text] to be [Formula: see text]-free. For [Formula: see text], we have shown that [Formula: see text] is a local ring, and the complete ideal structure of [Formula: see text] is determined. Some examples are presented.

scholarly journals Ideal structure of Zq + uZq and Zq + uZq-cyclic codes

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4199-4214
Author(s):  
Raj Kumar ◽  
Maheshanand Bhaintwal ◽  
Ramakrishna Bandi
Keyword(s):  
Cyclic Code ◽  
Cyclic Codes ◽  
Gray Map ◽  
Sufficient Condition ◽  
Ideal Structure ◽  
Necessary And Sufficient Condition ◽  
Complete Ideal ◽  
Necessary And Sufficient

In this paper, we study cyclic codes of length n over R = Zq + uZq, u2 = 0, where q is a power of a prime p and (n; p) = 1. We have determined the complete ideal structure of R. Using this, we have obtained the structure of cyclic codes and that of their duals through the factorization of xn-1 over R. We have also computed total number of cyclic codes of length n over R. A necessary and sufficient condition for a cyclic code over R to be self-dual is presented. We have presented a formula for the total number of self-dual cyclic codes of length n over R. A new Gray map from R to Z2rp is defined. Using Magma, some good cyclic codes of length 4 over Z9 + uZ9 are obtained.

ℤ2(ℤ2 + uℤ2)-Additive cyclic codes and their duals

2016 ◽  
Vol 08 (02) ◽  
pp. 1650027 ◽  
Author(s):  
B Srinivasulu ◽  
Maheshanand Bhaintwal
Keyword(s):  
Structural Properties ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Free Module ◽  
Necessary Condition ◽  
Spanning Set

In this paper, we study some structural properties of [Formula: see text]-additive cyclic codes in [Formula: see text] as [Formula: see text]-submodules of [Formula: see text], where [Formula: see text]. The generators for these codes are obtained and a minimal spanning set is determined for even and odd [Formula: see text] separately with arbitrary [Formula: see text]. We also determine the generators of duals of the [Formula: see text]-additive cyclic codes for odd [Formula: see text]. A necessary condition for a 1-generator [Formula: see text]-cyclic code to be a [Formula: see text]-free module is obtained.

On skew cyclic codes over a semi-local ring

2015 ◽  
Vol 07 (04) ◽  
pp. 1550042 ◽  
Author(s):  
Mohammad Ashraf ◽  
Ghulam Mohammad
Keyword(s):  
Local Ring ◽  
Structural Properties ◽  
Decomposition Method ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Gray Map ◽  
Gray Image ◽  
Skew Cyclic Codes

In the present paper, we study skew cyclic codes over the finite semi-local ring [Formula: see text], where [Formula: see text] and [Formula: see text] is an odd prime. We define a Gray map from [Formula: see text] to [Formula: see text] and investigate the structural properties of skew cyclic codes over [Formula: see text] using decomposition method. It is proved that the Gray image of a skew cyclic code of length [Formula: see text] over [Formula: see text] is a skew [Formula: see text]-quasi-cyclic code of length [Formula: see text] over [Formula: see text]. Further, it is shown that the skew cyclic codes over [Formula: see text] are principally generated.

Some results on the linear codes over the finite ring F2+v1F2+⋯+vrF2

2016 ◽  
Vol 14 (01) ◽  
pp. 1650012 ◽  
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis ◽  
Senol Eren
Keyword(s):  
Linear Codes ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Finite Ring ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Quantum Error ◽  
Quasi Cyclic Codes

In this paper, we study the structure of cyclic, quasi-cyclic codes and their skew codes over the finite ring [Formula: see text], [Formula: see text] for [Formula: see text]. The Gray images of cyclic, quasi-cyclic, skew cyclic, skew quasi-cyclic codes over [Formula: see text] are obtained. A necessary and sufficient condition for cyclic code over [Formula: see text] that contains its dual has been given. The parameters of quantum error correcting codes are obtained from cyclic codes over [Formula: see text].

On quantum codes via cyclic codes of arbitrary length over 𝔽4 + u𝔽4

2018 ◽  
Vol 10 (03) ◽  
pp. 1850033 ◽  
Author(s):  
Amit Sharma ◽  
Ramakrishna Bandi ◽  
Maheshanand Bhaintwal
Keyword(s):  
Linear Code ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Quantum Codes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Arbitrary Length ◽  
Gray Image ◽  
Necessary And Sufficient

In this paper, we study cyclic codes over [Formula: see text]. A necessary and sufficient condition for a cyclic code over [Formula: see text] to contain its dual is determined. The odd and even length cases are discussed separately to obtain above condition. It is shown that Gray image of a cyclic code over [Formula: see text] containing its dual is a linear code over [Formula: see text] which also contains its dual. We have then obtained the parameters of corresponding CSS-quantum codes over [Formula: see text]. By augmentation, we construct codes with dual-containing property from codes of smaller size containing their duals. Through this construction, we have obtained some optimal quantum codes over [Formula: see text]. Some examples have been given to illustrate the results.

On skew cyclic codes over a mixed alphabet and their applications to DNA codes

2021 ◽  
pp. 2150143
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis ◽  
Nuh Aydin
Keyword(s):  
Finite Field ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Dna Codes ◽  
Skew Cyclic Codes ◽  
Necessary And Sufficient

In this paper, we introduce skew cyclic codes over the mixed alphabet [Formula: see text], where [Formula: see text] is the finite field with 4 elements and [Formula: see text]. Our results include a description of the generator polynomials of such codes and a necessary and sufficient condition for an [Formula: see text]-skew cyclic code to be reversible complement.

Quasi-cyclic codes over the ring 𝔽p[u]/〈u2 − u〉

2015 ◽  
Vol 08 (04) ◽  
pp. 1550085
Author(s):  
Sukhamoy Pattanayak ◽  
Abhay Kumar Singh
Keyword(s):  
Structural Properties ◽  
Cyclic Codes ◽  
Natural Generalization ◽  
Spanning Set ◽  
The One ◽  
Quasi Cyclic Codes

Quasi-cyclic (QC) codes are a natural generalization of cyclic codes. In this paper, we study some structural properties of QC codes over [Formula: see text], where [Formula: see text] is a prime and [Formula: see text]. By exploring their structure, we determine the one generator QC codes over [Formula: see text] and the size by giving a minimal spanning set. We discuss some examples of QC codes of various length over [Formula: see text].

Frobenius closure and prime characteristic singularities

2020 ◽  
Author(s):  
◽  
Kyle Logan Maddox
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Sufficient Condition ◽  
Zariski Topology ◽  
Prime Characteristic ◽  
Joint Work ◽  
University Of Missouri ◽  
Mild Conditions ◽  
Flat Maps ◽  
The University

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] This dissertation outlines several results about prime characteristic singularities for which the nilpotent part under the induced Frobenius action on local cohomology is either finite colength or the entire module, collectively referred to here as nilpotent singularities. First, we establish a sufficient condition for the finiteness of the Frobenius test exponent for a local ring and apply it to conclude that nilpotent singularities have finite Frobenius test exponent. In joint work with Jennifer Kenkel, Thomas Polstra, and Austyn Simpson, we show that under mild conditions nilpotent singularities descend and ascend along faithfully flat maps. Consequently, we then prove that the loci of primes which are weakly F-nilpotent and F-nilpotent are open in the Zariski topology for rings which are either F-finite or essentially of fiiite type over an excellent local ring.

Separability criteria based on the Bloch representation of density matrices

2007 ◽  
Vol 7 (7) ◽  
pp. 624-638
Author(s):  
J. de Vicente
Keyword(s):  
Sufficient Conditions ◽  
Quantum Systems ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Pure Product ◽  
Separability Problem ◽  
Arbitrary Dimensions ◽  
Necessary And Sufficient ◽  
Bloch Representation

We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.

Rising or Constrained Power?

2010 ◽  
Author(s):  
Eswaran Sridharan
Keyword(s):  
Economic Growth ◽  
High Growth ◽  
The United States ◽  
Asia Pacific ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Economic Weight ◽  
Rising Power ◽  
Global Influence

This chapter analyses India’s prospects as a rising power by asking what kind of power India has the potential to be, given its military, economic, and institutional capacities and the economic and geostrategic constraints it faces. It argues that while sustained high growth is a necessary condition it is not a sufficient condition since economic growth does not necessarily convert smoothly into greater power. Due to such conversion problems India, like some other powers, might not be able to exercise commensurate regional, extra-regional, and global influence as might appear to follow from the revival of sustained high growth and increased economic weight. The more achievable and likely alternative is that of a coalitional or bridging power that can play the role of an effective partner in the security and other spheres to a range of powers, principally to the United States and in the Asia-Pacific and Indian Ocean regions.

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