BASES AND DIMENSION OF TWISTED CENTRALIZER CODES

• Main Author: Pradananta, Galih
• Format: Theses
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/39184
• Institution: Institut Teknologi Bandung

In 2017, Adel Alahmadi and his coauthors introduced the twisted centralizer code at their article [2]. Let A be a nn matrix over some finite field F and be element in F. The centralizer of A, twisted by defined to be fX 2 Fnn j AX = XAg and is denoted by CF (A; ). Almost in any situation, if there is no ambiguation about the field, we will write CF (A; ) as C(A; ). We can show that C(A; ) is a linear subspace of Fnn. Then it is a linear code, by reading them (as codeword) column-by-column. Eventually, it is so-called by the centralizer code of A and twisted by [2]. Let F[x] be polynomial ring over field F. Now we define an action f(x):u = f(A)u for every f(x) 2 F[x] and u 2 Fn. With this action, C(A; ) is F[x]-module. It is called by F[x]-module which induced by A. Three years before, in 2014, Adel Alahmadi and his coauthors introduced the socalled centralizer code [1]. Actually, twisted centralizer code is a generalization of centralizer code. We can see centralizer code as twisted centralizer code then twisted by 1. As F[x]-modul , we can examine centralizer code equal to F[x]- modul endomorphisms collection in Fn [8]. From the idea in [8], we show that twisted centralizer code is collection of F[x]-module homomorphisms. This insight gives us a way to study the structure of twisted centtralizers code and its properties, especially about its bases and dimension. We all understand that linear code has 3 parameters [n; k; d] with n=length of codewords, k=its dimension, and d=closest distances between any codewords. Unfortunately at Adel Alahmadi et.al. articles [1] and [2], there is not a general result about dimension of centralizer code nor twisted centralizer code. The most advanced result about its dimension is explained in Adel Alahmadi et.al. another article [3]. But it is only stated exact dimension at special cases that are when A is diagonalizable and A is cyclic. In this thesis, we show that twisted centralizer code is collection of F[x]-module homomorphism. So we can see that twisted centralizer code itself is F[x]-module. Then we can decompose it by having observation on primary cyclic decomposition iii of Fn. Finally, its bases and dimension can be determined.