On Quasi-Cyclic Codes as a Generalization of Cyclic Codes

In this article we see quasi-cyclic codes as block cyclic codes. We generalize some properties of cyclic codes to quasi-cyclic ones such as generator polynomials and ideals. Indeed we show a one-to-one correspondence between l-quasi-cyclic codes of length m and ideals of M_l(Fq)[X]/(X^m-1). This permits to construct new classes of codes, namely quasi-BCH and quasi-evaluation codes. We study the parameters of such codes and propose a decoding algorithm up to half the designed minimum distance. We even found one new quasi-cyclic code with better parameters than known [189, 11, 125]_F4 and 48 derivated codes beating the known bounds as well.Comment: (18/08/2011

Matrix Product Structure of a Permuted Quasi Cyclic Code and Its dual

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

Asymptotically Good Additive Cyclic Codes Exist

Long quasi-cyclic codes of any fixed index $>1$ have been shown to be asymptotically good, depending on Artin primitive root conjecture in (A. Alahmadi, C. G\"uneri, H. Shoaib, P. Sol\'e, 2017). We use this recent result to construct good long additive cyclic codes on any extension of fixed degree of the base field. Similarly self-dual double circulant codes, and self-dual four circulant codes, have been shown to be good, also depending on Artin primitive root conjecture in (A. Alahmadi, F. \"Ozdemir, P. Sol\'e, 2017) and ( M. Shi, H. Zhu, P. Sol\'e, 2017) respectively. Building on these recent results, we can show that long cyclic codes are good over \F_q, for many classes of $q$'s. This is a partial solution to a fifty year old open problem