M algebraic-coding-theory-notes.bib → algebraic-coding-theory-notes.bib

@@ -1,6 +1,6 @@
 %% LaTeX2e file `algebraic-coding-theory-notes.bib'
 %% generated by the `filecontents' environment
-%% from source `algebraic-coding-theory-notes' on 2026/03/08.
+%% from source `algebraic-coding-theory-notes' on 2026/03/21.
 %%
 @misc{hill,
   author = {Raymond Hill},

M algebraic-coding-theory-notes.pdf → algebraic-coding-theory-notes.pdf

M algebraic-coding-theory-notes.tex → algebraic-coding-theory-notes.tex

@@ -368,6 +368,92 @@ for the code of \ref{h.6.5},
   $$P_{undetec}(C) = \underbrace{1 \cdot p^2 (1-p)^2}_{0101} + \underbrace{2 \cdot p^3(1-p)^{4-3=1} }_{1011,~1110}= p^2 -p^4$$
 \end{eg}
 
+\subsection{Exercises}
+\begin{ex}{5.1}
+  is the binary (11, 24, 5)-code linear?
+\end{ex}
+\begin{proof}
+  recall: a q-ary linear $[n,k,d]$-code is also a q-ary $(n, q^k, d)$-code, but
+  the opposite is not always true.
+
+  $\Longrightarrow~ (11, 24, 5)$-code to be linear it would mean that $24=q^k$.
+
+  Since it is binary, $q=2$. But $24 \neq 2^k$, so it can not be linear.
+\end{proof}
+
+\begin{ex}{5.2}
+  $E_n$ is the code of all even-weight vectors of $V(n,2)$, which is linear.
+  What are the parameters $[n, k, d]$ of $E_n$? Write a GM for $E_n$ in standard
+  form.
+\end{ex}
+\begin{proof}
+  $$E_n = \{ x \in V(n, 2) | w(x) ~\text{is even} \}$$
+
+  $\longrightarrow$ is the binary even-weight ode (the single parity-check code);\\
+  it's linear since
+  \begin{itemize}
+    \item the zero vector has even weight
+    \item over $GF(2)$, the sum of two even-weight vectors has even weight
+  \end{itemize}
+
+  Recall: $F_q^n \Longleftrightarrow \Vnq$, so $V(n, 2) \cong F_2^n$
+
+  $$\Longrightarrow~ E_n = \{ (x_1, \ldots, x_n) \in F_2^n | x_1 + \ldots + x_n = 0 \}$$
+
+  minimum distance is $2$, since
+  \begin{itemize}
+    \item no word of weight 1 is in $E_n$
+    \item words of weight 2 are in $E_n$
+  \end{itemize}
+
+  Therefore, parameters are $[n, n-1, 2]$
+
+  GM:
+  $$G=\left[ I_{n-1} | 1 \right]$$
+
+  which generates vectors of the form
+  $$(a_1, \ldots, a_{n-1}, a_1+\ldots+a_{n-1})$
+\end{proof}
+
+\begin{ex}{5.5}
+  Prove that in a \emph{binary linear code}, either all the codewords have even weight or exactly half have even weight and half odd weight.
+\end{ex}
+\begin{proof}
+  let $C_e,~ C_o$ be subsets of $C$ with even and odd weights respectively.
+
+  So that $C = C_e \cup C_o$ (disjoint union).
+
+  For binary vectors $x,y$:
+  $$w(x+y) = w(x) + w(y) \pmod 2$$
+
+  ie. over $F_2$, the parity of the weight is additive:
+  \begin{align*}
+    even + even &= even\\
+    even + odd &= odd\\
+    odd + odd &= even
+  \end{align*}
+
+  Now we have two possible cases:
+  \begin{itemize}
+    \item $C$ has no odd weighted codewords:\\
+      $C_0 = \emptyset,~ \forall x \in C$ $x$ is even weighted
+    \item $C$ has at least one odd-weighted codeword:\\
+      let $u \in C_o$, consider the map
+      \begin{align*}
+        \phi: C_e &\longrightarrow C_o\\
+        c &\longmapsto c+u
+      \end{align*}
+      with $c \in C_e,~ u \in C_o,~ \phi(c) \in C_o$.
+
+      $\phi$ is bijective, since $\phi(\phi(c))=(c+u)+u=c$,
+
+      so $\phi$ is it's own inverse; hence $|C_e| = |C_o|$.
+
+      Since $C=C_e \cup C_e$, half codewords are even weighted and half odd.
+  \end{itemize}
+
+\end{proof}
+
 \bibliographystyle{unsrt}
 \bibliography{algebraic-coding-theory-notes.bib}