Multidimensional quasi-cyclic and convolutional codes
Özkaya, Buket (2014) Multidimensional quasi-cyclic and convolutional codes. [Thesis]
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 eld. 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 su cient condition for noncatastrophic nD polynomial encoders is obtained for 1-generator nD convolutional codes.
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