BASES AND DIMENSION OF TWISTED CENTRALIZER CODES
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 situati...
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| Format: | Theses |
| Language: | Indonesia |
| Online Access: | https://digilib.itb.ac.id/gdl/view/39184 |
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| Institution: | Institut Teknologi Bandung |
| Language: | Indonesia |
| Summary: | 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. |
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