Artin–Schreier curves and weights of two-dimensional cyclic codes

Güneri̇, Cem

doi:10.1016/j.ffa.2003.10.002

Cited by 22 publications

(19 citation statements)

References 11 publications

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“…Since τ vij necessarily divides m, R is a subring of R. Therefore T r R/ R makes sense. From a j ∈ R \ {0}, we have |U vij | = m. By Theorem 2.2 in [12], we deduce that there are q (m−τvi j )s a j 's in R such that T r R/R a j ξ vij = 0. The number of elements in the kernel of T r R/ R is also…”

Section: Cyclic Codesmentioning

confidence: 86%

Bounds on quasi-cyclic codes over finite chain rings

Gao

Shen

2015

J. Appl. Math. Comput.

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In this paper, we mainly consider quasi-cyclic (QC) codes over finite chain rings. We study module structures and trace representations of QC codes, which lead to some lower bounds on the minimum Hamming distance of QC codes. Moreover, we investigate the structural properties of 1-generator QC codes. Under some conditions, we discuss the enumeration of 1-generator QC codes and describe how to obtain the one and only one generator for each 1-generator QC code.

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Section: Cyclic Codesmentioning

confidence: 86%

Bounds on quasi-cyclic codes over finite chain rings

Gao

Shen

2015

J. Appl. Math. Comput.

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“…Let us note that the number of rational points on Artin-Schreier type hypersurfaces over finite fields helps us estimate the minimum distance of multidimensional cyclic codes via the trace representation of this class of codes (see [13,15], and also [32] for another relation between algebraic geometry and multidimensional cyclic codes). Multidimensional cyclic codes are closely related to multidimensional QC codes, as we will explain in this thesis.…”

Section: Multidimensional Versions Of Convolutional Codes Have Been Smentioning

confidence: 99%

“…So, an analysis similar to those in [13,15] can be in principal applied to QnDC codes and the relation with certain nD convolutional codes can be used to write a lower bound on the free distance of nD convolutional codes in terms of rational points on Artin-Schreier hypersurfaces. This remains as a work to be done in the future.…”

Section: Multidimensional Versions Of Convolutional Codes Have Been Smentioning

confidence: 99%

Multidimensional Quasi-Cyclic and Convolutional Codes

Güneri̇

Özkaya

2016

IEEE Trans. Inform. Theory

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We introduce multidimensional generalizations of quasi-cyclic codes and investigate their algebraic properties as well as their links to multidimensional convolutional codes. We call these generalized codes n-dimensional quasi-cyclic (QnDC) codes. We provide a concatenated structure for QnDC codes in the sense that they can be decomposed into shorter codes over extensions of their base field. This structure allows us to prove that these codes are asymptotically good. Then, we extend the relation between quasi-cyclic and convolutional codes to multidimensional case. Lally has shown that the free distance of a convolutional code is lower bounded by the minimum distance of an associated quasi-cyclic code. We show that a QnDC code can be associated to a given nD convolutional code. Moreover, we prove that the relation between distances of convolutional and quasicyclic codes extend to a class of 1-generator 2D convolutional codes and the associated Q2DC codes. Along the way, an alternative new description of noncatastrophic polynomial encoders is given for 1-generator 1D convolutional codes and a sufficient condition for noncatastrophic nD polynomial encoders is obtained for 1-generator nD convolutional codes.

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“…where δ is the least common multiple of the degrees of α 1 i and α 2 j over F q . We can write Ω as a disjoint union of these F q -conjugacy classes [4].…”

Section: Two Dimensional (2-d) Cyclic Codesmentioning

confidence: 99%

Quantum Stabilizer Codes

Gaitan¹

2008

Quantum Error Correction and Fault Tolerant Quantum Computing

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This thesis would not have been possible without the support of many people. First and foremost I would like to express my deepest gratitude to my advisor Dr. Cem Güneri whose guidance, support and assistance from the initial to the final phase has enabled me to successfully develop this thesis. I would like to thank my committee members, Dr. Kagan Kurşungöz and Dr. Seher Tutdere who have offered their assistance throughout this period. Finally, I would like to thank my parents, who were not here with me throughout my studies, but they motivated me to pursue the master degree and I constantly received their love and support. Last but not the least, I would like to thank my wife, Zohreh for her love and unyielding support. Thank you god for giving me all the strength that I needed. viii

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Artin–Schreier curves and weights of two-dimensional cyclic codes

Cited by 22 publications

References 11 publications

Bounds on quasi-cyclic codes over finite chain rings

Bounds on quasi-cyclic codes over finite chain rings

Multidimensional Quasi-Cyclic and Convolutional Codes

Quantum Stabilizer Codes

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