Modul: MAT076 Arbeitsgemeinschaft in Codierungstheorie und Kryptographie

## On the Algebraic Structure of Quasi-Cyclic Codes

Dr. Elif Sacikara talk

Speaker invited by: Prof. Dr. Joachim Rosenthal

**Date:** 28.11.18 **Time:** 16.00 - 17.00 **Room:** Y27H12

Quasi-cyclic (QC) codes have been investigated for more than 50 years and they play an important role not only in algebraic coding theory but also in code-based cryptography. These codes belong to the family of linear block codes and can be described both in an algebraic and a vectorial way. In this talk, we will mostly concentrate on the algebraic structure, and view a QC code of index l as an R-submodule in R^l for positive integers l, m and the quotient R = F_q[x]/(x^m − 1). The special case l = 1 gives an ideal in R, which is a cyclic code of length m. So, QC codes are a natural generalization of cyclic codes. In the first part of the talk, in order to explain the algebraic structure, we will present the technique of Groebner bases for modules, applied by Lally and Fitzpatrick to the family of QC codes. Then we will view another algebraic approach to QC codes called the Chinese Remainder Theorem (CRT) decomposition, introduced by Ling and Sole, that allows us to consider a QC code as a direct sum of certain linear codes of shorter lengths. We will also present the concatenated structure of QC codes, described by Jensen, and observe the equivalence of the CRT decomposition and the concatenated structure of a QC code, proven by Guneri and Ozbudak. Finally, we will briefly overview the relations between QC codes and other code families.