## Dissertations

Dissertation

#### Degree

Doctor of Philosophy

Mathematics

#### Date of Defense

4-16-2021

Ravindra Girivaru

#### Committee

Ravindra Girivaru

Prabhakar Rao

David Covert

#### Abstract

In my Dissertation I will work mostly with Permuted Quasi Cyclic Codes. They are a generalization of Cyclic Codes, one of the most important families of Linear Codes in Coding Theory. Linear Codes are very useful in error detection and correction. Error Detection and Correction is a technique that first detects the corrupted data sent from some transmitter over unreliable communication channels and then corrects the errors and reconstructs the original data. Unlike linear codes, cyclic codes are used to correct errors where the pattern is not clear and the error occurs in a short segment of the message.

The length of Permuted Cyclic Codes usually is a big number, that is why I will try to break them down into cyclic codes of small length. This way we can make the study of these code easier and understand them better.

One way of breaking down big codes is to write them down as matrix product of small codes. From any permuted quasi cyclic code, we can define some special cyclic codes. I will try to find a sufficient and necessary conditions so any permuted quasi cyclic code can be written as a matrix product of those codes.

Another generalization of cyclic codes is the family of multi cyclic codes. These types of

codes are more complicated than the previous one so I will propose to limit myself on finding the

structure of ternary multi cyclic codes of length 4.

One technique of constructing new linear codes from a given linear code is by finding the so

called Euclidean dual of a linear code. In my thesis I will also analyze the Euclidean dual of the

families above.

COinS