Skew cyclic codes over Fq + uFq + vFq

2018 ◽  
Vol 11 (05) ◽  
pp. 1850072 ◽  
Author(s):  
Mohammad Ashraf ◽  
Ghulam Mohammad
Keyword(s):  
Structural Properties ◽  
Decomposition Method ◽  
Cyclic Codes ◽  
Gray Map ◽  
Skew Cyclic Codes ◽  
Quasi Cyclic Codes

In this paper, we study skew cyclic codes over the ring [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is a 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 shown that the Gray images of skew cyclic codes of length [Formula: see text] over [Formula: see text] are the skew [Formula: see text]-quasi cyclic codes of length [Formula: see text] over [Formula: see text]. Finally, the idempotent generators of skew cyclic codes over [Formula: see text] have also been discussed.

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.

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].

scholarly journals Skew cyclic codes over 𝔽4R

2020 ◽  
pp. 2250065
Author(s):  
Nasreddine Benbelkacem ◽  
Martianus Frederic Ezerman ◽  
Taher Abualrub ◽  
Nuh Aydin ◽  
Aicha Batoul
Keyword(s):  
Error Control ◽  
Cyclic Codes ◽  
Gray Map ◽  
Inner Product ◽  
Algebraic Structures ◽  
Error Control Codes ◽  
Dna Codes ◽  
Natural Connection ◽  
Skew Cyclic Codes

This paper considers a new alphabet set, which is a ring that we call [Formula: see text], to construct linear error-control codes. Skew cyclic codes over this ring are then investigated in details. We define a nondegenerate inner product and provide a criteria to test for self-orthogonality. Results on the algebraic structures lead us to characterize [Formula: see text]-skew cyclic codes. Interesting connections between the image of such codes under the Gray map to linear cyclic and skew-cyclic codes over [Formula: see text] are shown. These allow us to learn about the relative dimension and distance profile of the resulting codes. Our setup provides a natural connection to DNA codes where additional biomolecular constraints must be incorporated into the design. We present a characterization of [Formula: see text]-skew cyclic codes which are reversible complement.

Skew quasi cyclic codes over 𝔽q + v𝔽q

2019 ◽  
Vol 18 (04) ◽  
pp. 1950077 ◽  
Author(s):  
Mehmet Özen ◽  
N. Tuğba Özzaim ◽  
Halit İnce
Keyword(s):  
Cyclic Codes ◽  
Quantum Codes ◽  
Computer Search ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generating Set ◽  
Skew Cyclic Codes ◽  
Necessary And Sufficient ◽  
Quantum Parameters ◽  
Quasi Cyclic Codes

In this work, skew quasi cyclic codes over [Formula: see text], where [Formula: see text] are considered. The generating set for one generator skew quasi cyclic codes over [Formula: see text] is also determined. We discuss a sufficient condition for one generator skew quasi cyclic codes to be free. Furthermore, a BCH type bound is given for free one generator skew quasi cyclic codes. We investigate the dual of skew quasi cyclic codes over [Formula: see text]. We give a necessary and sufficient condition for skew cyclic codes over [Formula: see text] to contain its dual. Moreover, we construct quantum codes from skew cyclic codes over [Formula: see text]. By using computer search we give some examples about skew quasi cyclic codes and list some quantum parameters in the table.

scholarly journals The homogeneous weight for Rk, related gray map and new binary quasi-cyclic codes

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 885-897 ◽  
Author(s):  
Bahattin Yildiz ◽  
Ismail Kelebek
Keyword(s):  
Coding Theory ◽  
Cyclic Codes ◽  
Gray Map ◽  
Binary Codes ◽  
Binary Images ◽  
Frobenius Rings ◽  
Homogeneous Weights ◽  
Homogeneous Weight ◽  
Theoretical Results ◽  
Quasi Cyclic Codes

Using theoretical results about the homogeneous weights for Frobenius rings, we describe the homogeneous weight for the ring family Rk, a recently introduced family of Frobenius rings which have been used extensively in coding theory. We find an associated Gray map for the homogeneous weight using first order Reed-Muller codes and we describe some of the general properties of the images of codes over Rk under this Gray map. We then discuss quasi-twisted codes over Rk and their binary images under the homogeneous Gray map. In this way, we find many optimal binary codes which are self-orthogonal and quasi-cyclic. In particular, we find a substantial number of optimal binary codes that are quasi-cyclic of index 8, 16 and 24, nearly all of which are new additions to the database of quasi-cyclic codes kept by Chen.

On the linear codes over the ring Rp

2016 ◽  
Vol 08 (02) ◽  
pp. 1650036 ◽  
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis ◽  
Senol Eren
Keyword(s):  
Prime Number ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Constacyclic Codes ◽  
Nontrivial Automorphism ◽  
Macwilliams Identities ◽  
Skew Cyclic Codes ◽  
Quantum Error ◽  
Quasi Cyclic Codes

Some results are generalized on linear codes over [Formula: see text] in [15] to the ring [Formula: see text], where [Formula: see text] is an odd prime number. The Gray images of cyclic and quasi-cyclic codes over [Formula: see text] are obtained. The parameters of quantum error correcting codes are obtained from negacyclic codes over [Formula: see text]. A nontrivial automorphism [Formula: see text] on the ring [Formula: see text] is determined. By using this, the skew cyclic, skew quasi-cyclic, skew constacyclic codes over [Formula: see text] are introduced. The number of distinct skew cyclic codes over [Formula: see text] is given. The Gray images of skew codes over [Formula: see text] are obtained. The quasi-constacyclic and skew quasi-constacyclic codes over [Formula: see text] are introduced. MacWilliams identities of linear codes over [Formula: see text] are given.

Optimal quantum codes from additive skew cyclic codes

2016 ◽  
Vol 08 (03) ◽  
pp. 1650037 ◽  
Author(s):  
Nuh Aydin ◽  
Taher Abualrub
Keyword(s):  
Structural Properties ◽  
Cyclic Codes ◽  
The Other ◽  
Quantum Codes ◽  
Additive Codes ◽  
Other Hand ◽  
Skew Cyclic Codes

Additive codes received much attention due to their connections with quantum codes. On the other hand, skew cyclic codes proved to be a useful class of codes that contain many good codes. In this work, we introduce and study additive skew cyclic codes over the quaternary field [Formula: see text], obtaining some structural properties of these codes. Moreover, we also show that many best known and optimal quantum codes can be obtained from this class.

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