deletions | additions
diff --git a/bits3.tex b/bits3.tex
index 7de4fd1..542d3b9 100644
--- a/bits3.tex
+++ b/bits3.tex
...
Another structure useful to manage ordered sets, like vectors, is the set of
polynomials described in
section Section \ref{Sec:Polynomials}.
We recall that the sum of two polynomials, $x+1, x^2+1 \in \Fb$, is %that is $$(x+1) + (x^2+1)= x^2 + x+(1+1).$$ Since in $\Fb$ $1+1=0$, we finally get
\begin{equation}\label{eq:pol}
(x+1) + (x^2+1)= x^2+x.
\end{equation}
\begin{Exercise}\label{Vectors2Poly}
Do you see any connections between
\ref{eq:vec} (\ref{eq:vec}) and
\ref{eq:pol}? (\ref{eq:pol})?
\end{Exercise}
As you have seen in the previous exercise, nice and meaningful operations on vectors can be interpreted as operations on polynomials. In the rest of the section we give other examples.
...
\begin{equation}\label{shiftpol}
x^4+x^3+x {\Longleftrightarrow} x^3+x^2+1.
\end{equation}
That
is is, shifting to the left a
vectors corresponds vector seems to correspond to
multiply the
multiplication of its polynomial by
$x$ and $x$, while shifting to the right
is dividing seems to correspond to the
division of its polynomial by
$x$ itself. $x$.
\begin{Exercise}
Suppose you want to
make double a
copy of a vectors vector of $n$ bits. For example let $n=3$
and we want that $(1,0,1)$ becomes $(1,0,1,1,0,1)$. Using the polynomial
notation for the vector and multiplying by
a new polyomial another polynomial $p$ you can obtain
the result expected. Who is $p$?
\end{Exercise}
Another example that is of big interest is the following. Suppose that the fourth bit (the one with ``?'' symbol) in a vector
$x$ $v$
\[
x=(?,1,1,0) v=(?,1,1,0)
\]
is computed by the sum of the first and the third. In languages as ``C'' we can say
\[
x[3]=x[2]+x[0] v[3]=v[2]+v[0];
\]
where
$x[0]$ $v[0]$ represents the first entry of the vector and
$x[2]$ $v[2]$ the third one. In a polynomial notation this ``function'' becomes
\[
x^3=x^2+x^0
\]
where that is with the exponents
represent representing the positions of the
entries of the vector. vector entries.
...
%\end{Exercise}
Let us consider the vector $(0,1,1)\in (\Fb)^3$, associated to the polynomial $f=x^3+x+1$ as explained above.
We
point out observe that this polynomial is irreducible.
We can construct a Linear Feedback Shift Register (LFSR) over three bits, using $f=x^3+x+1$.
\\
