Tagged Questions
2
votes
2answers
132 views
The linearity of the extended code
If we add a last digit to the code $C$ of length $n$, we obtain a new code called extended code. My question is:
If the code $C$ is linear, can we prove that the extended code $C'$ is linear too?
...
1
vote
1answer
54 views
extended linear codes over the field $\mathbb F_q$
Suppose we extend the $[n,k]$ linear code $C$ over the field $\Bbb F_q$ to the code $C'$, where
$$
C' = \{(x_1,\ldots ,x_n,x_{n+1})\in \Bbb F_q^{n+1} : (x_1,\ldots,x_n) \in C \text{ and } ...
7
votes
0answers
102 views
Error correction code handling deletions and insertions
I have data which is expressed in form of fixed-length sequence of decimal digits, typically 10 digits in length.
I'm looking for error correction code that would allow me to append one or more ...
1
vote
1answer
91 views
Field construction
Explain how to construct a field of order $343$ not using addition and multiplication tables.
I understand that every finite field has order $p^n$ for some prime $p$. Since $343$ is $7^3$, let ...
4
votes
6answers
216 views
How to construct minimal polynomial?
This is an exam question from last semester.
We have the finite field
$$ \mathbb F_{81} = \mathbb Z_3 [x]/(x^4+x^2+x+1)$$
(a) Prove that the polynomial $$ x^4+x^2+x+1 $$
is irreducible
(b) ...
4
votes
2answers
138 views
How to determine if this is a field?
A finite subring $R$ of a field $V$ contains $1$ (so $1$ is an element of $R$).
The question is:
True or False: The ring $R$ must be a field.
I thought that if $R$ was a field it had to be a finite ...
