\(b\)-Symbol Weight Distribution of Irreducible Cyclic Codes
Talk by Blerim Alimehaj
Date: 18.12.24 Time: 15.15 - 16.00 Room: Y27H28
As our technology advances, the need for \(b\)-symbol read channels that can handle messages with high-density data becomes crucial. The problem with conventional read channels is that they are more likely to overlap multiple information units, also called symbols, while reading messages with high-density data. The idea behind \(b\)-symbol read channels is that these channels consider all \(b\) consecutive symbols from the sent message as one symbol. This protects the message from being read with overlapping symbols. Considering a code word of some code \(\mathcal{C}\) as the sent message, the message read by some \(b\)-symbol read channel is called a \(b\)-symbol code word. In this thesis, we investigate \(b\)-symbol codes over semiprimitive irreducible cyclic codes and their Hamming weight distribution. The \(b\)-symbol Hamming weight distribution of semiprimitive irreducible cyclic codes is determined up to an invariant that we call \(\mu(b)\) and \(\mu_l (b)\). These invariants depend on \(b\) and on the choice of the primitive element that we use to describe the irreducible cyclic code\(\mathcal{C}\). This thesis aims to find lower and upper bounds of the average values of these invariants. To do so, we use algebraic function field theory and number theory. In this thesis, we obtain several lower and upper bounds, test their performance for smaller fields, and compare them. Furthermore, we are able to improve those bounds due to a number theoretic approach. Using these bounds, we are able to deduce a very good estimate for the average values of the invariants \(\mu(b)\) resp. \(\mu_l (b)\). These results provide a better understanding of the \(b\)-symbol Hamming weight distribution of semiprimitive irreducible cyclic codes.