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

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

@@ -1702,6 +1702,7 @@
 \vspace{1.0cm}
 
 \section{Cyclic Codes}
+This section contains notes both from Hill \cite{hill} and Lint \cite{lint} books. Hill's notes are enumerated as $X.Y$ while Lint's notes as $X.Y.Z$.
 
 \subsection{Recap on ideals}
 More about ideals at the \emph{Commutative Algebra notes} \cite{commutativealgebranotes}.
@@ -1837,6 +1838,45 @@ \item Every ideal in $R_n$ is a \emph{principal ideal}.\\
       ie. $R_n$ is a PID.
   \end{enumerate}
 \end{remark}
+
+\vspace{0.5cm}
+\begin{thm}{6.1.3}
+  a linear code $C$ in $\mathbb{F}_q^n$ is cyclic iff $C$ is an ideal in $\mathbb{F}_q[x]/(x^n -1)$.
+\end{thm}
+\begin{proof}
+  If $C$ ideal in $\mathbb{F}[x]/(x^n -1)$ and $c(x)=c_0 + c_1 x + \ldots + c_{n-1}x^{n-1} \in C$, then
+  \begin{align*}
+    x \cdot c(x) &= c_0 x + c_1 x \cdot x + \ldots + \underbrace{c_{n-1} x^{n-1} \cdot x}_{=c_{n-1} \cdot 1 \pmod{x^n-1}}\\
+                 &= c_{n-1} + c_0 x + c_1 x^2 + \ldots + c_{n-2} x^{n-1} \in C\\
+    \Longrightarrow&~ (c_{n-1}, c_0, c_1, \ldots, c_{n-2}) \in C
+  \end{align*}
+
+  \vspace{0.5cm}
+  Conversely, if $C$ cyclic, then $\forall c(x) \in C,~ x \cdot c(x) \in C$.
+
+  Therefore $x^i \cdot c(x) \in C ~\forall i \in [n]$
+
+  and since $C$ is linear, $a(x) \cdot c(x) \in C$
+
+  ie.
+  $$a_0 \cdot c(x) + a_1 \cdot x \cdot c(x) + a_2 \cdot x^2 \cdot c(x) + \ldots + a_{n-1} \cdot x^{n-1} \cdot c(x) \in C$$
+  $\forall a(x) \in \mathbb{F}_q[x]/(x^n-1)$.
+\end{proof}
+
+\begin{cor}{}
+  $\matbb{F}_q[x]/(x^n -1)$ is a PID (Principal Ideal Domain).
+
+  Thus $C$ consists of the multiples of $g(x) \in C$ monic and of lowest degree.
+
+  Therefore,
+  \begin{itemize}
+    \item[$\Longrightarrow$] $g(x)$ is the \emph{generator polynomial} (more at \ref{12.9}) of $C$: $C= \langle g(x) \rangle$.
+    \item[$\Longrightarrow$] $g(x)$ is divisor of $x^n -1$, ie. $g(x) | (x^n-1)$.\\
+      since otherwise $gcd(x^n-1, g(x))$ would be a polynomial in $C$ with degree $< deg(g(x))$, a contradiction.
+
+  \end{itemize}
+\end{cor}
+\vspace{0.5cm}
 
 \vspace{1.0cm}
 Let $f(x) \in R_n$, $\langle f(x) \rangle$ the subset of $R_n$ consisting of all multiples of $f(x)$ (reduced modulo $x^n-1$). ie.
@@ -2029,6 +2069,124 @@ \end{align*}
 
   since $C$ is an ideal of $R_n$ and $g(x) \in C$ and $q(x) \in R_n$.
 \end{proof}
+
+\subsection{Maximal and minimal cyclic codes}
+
+Let $x^n-1 = f_1(x) \cdot f_2(x) \cdot \ldots \cdot f_t(x)$, with $f_i(x)$ irreducible polynomials.
+
+\begin{defn}{6.1.6}[maximal and minimal cyclic codes]
+  the cyclic code generated by $f_i(x)$ is called a \emph{maximal cyclic code} (since it's a maximal ideal), denoted $M_i^+$.
+
+  The code generated by $(x^n-1)/f_i(x)$ is called a \emph{minimal cyclic code}, denoted $M_i^-$. Also named \emph{irreducible code}.
+\end{defn}
+
+\vspace{0.5cm}
+
+\begin{defn}{}[Zeros of a cyclic code]
+  Let $x^n-1= f_1(x) \cdot \ldots \cdot f_t(x)$. Let $\beta_i$ be a zero of $f_i(x)$ in some extension field of $\mathbb{F}_q$\\
+  ie. $f_i(\beta)=0$, and $f_i(x)$ is the minimal polynomial of $\beta_i$.
+\end{defn}
+
+$\Longrightarrow~ M_i^+$ is the set of polynomials $c(x)$ for which $c(\beta_i)=0$.
+
+In general, a cyclic code can be specified by requiring that all codewords have certain zeros: require that all codewords have $\beta_i$ points as zeros. For example:
+
+Start by any set $\alpha_1, \alpha_2, \ldots, \alpha_s$. Define the code $C$ by
+$$c(x) \in C ~\text{iff}~ c(\alpha_i)=0 ~\forall i=1,2, \ldots, s$$
+then $C$ is cyclic, and the generator polynomial is the least common multiple (lcm) of the minimal polynomials of $\alpha_1, \alpha_2, \ldots, \alpha_s$.
+
+\subsection{Hamming cyclic codes}
+\begin{thm}{6.3.1}
+  let $n := (q^m - 1)/(q-1)$, let $\beta$ a primitive $n$-th root of unity in $\mathbb{F}_{q^m}$. Let $(m, q-1)=1$.
+
+  The cyclic code $C:= \{ c(x) | c(\beta)=0 \}$ is equivalent to the $[, n-m]$ Hamming code over $\mathbb{F}_q$.
+\end{thm}
+\begin{proof}
+  since $n=(q-1) \cdot (q^{m-2} + 2 q^{m-3} + \ldots + m-1) + m$ have $(n, q-1) = (m, q-1) = 1$.
+
+  Therefore $\beta^{i(q-1)} \neq 1 ~\forall i \in [n]$, ie. $\beta^i \neq \mathbb{F}_q ~\forall i \in [n]$.
+
+  Columns of $H$, which are the representations of $1, \beta, \beta^2, \ldots, \beta^{n-1}$ as vectors in $\mathbb{F}_{q^m}$, are pairwise linearly independent over $\mathbb{F}_q$.
+
+  So $H$ is the PCM of an $[n, n-m]$ Hamming code.
+\end{proof}
+
+\begin{eg}{6.3.2}
+  smallest extension of $\mathbb{F}_2$ which contains a primitive 9-th root of unity is $\mathbb{F}_{2^6}$.
+
+  Let $\alpha$ a primitive element of the field, then $\alpha^63=1$ ,and $\beta := \alpha^7$ is a primitive 9-th root of unity.
+
+  Then, the minimal polynomial of $\beta$ has the zeros $\beta, \beta^2, \beta^4, \beta^8, \beta^{16}=\beta^7, \beta^{14}=\beta^5$.
+
+  $\Longrightarrow$ min. polynomial: $(x^9 - 1)/(x^3 -1) = x^6 + x^3 + 1$.
+
+  So $(x^9-1) = \underbrace{(x-1)}_{f_1(x)} \underbrace{(x^2+x+1)}_{f_2(x)} \underbrace{(x^6+x^3 + 1)}_{f_3(x)}$.
+
+  The code $M_3^+$ has pairwise independent columns in $H$, ie. min.distance $\geq 3$.
+
+  $M_3^+$ consists of codewords $(c_0 c_1 c_2 ~~ c_0 c_1 c_2 ~~ c_0 c_1 c_2)$, so $d=3$.
+
+  The code $M_3^-$ has check polynomial $x^6 + x^3 + 1$, so it's a $[9, 6]$-code.
+
+  Since $x^3-1 \in M_3^-$ (is a codeword), the distance is $2$.
+\end{eg}
+
+
+\subsection{Idempotent of a cyclic code}
+
+\begin{thm}{6.4.1}
+  $\exists!~ c(x) \in C$ which is an identity element for $C$.
+\end{thm}
+\begin{proof}
+  let $g(x)$ generator polynomial, $h(x)$ check polynomial.
+
+  ie. $g(x) \cdot h(x) = x^n -1$ in $\mathbb{F}_q[x]$.
+
+  Since $x^n -1$ has no multiple zeros (ie. has simple zeros), we have $(g(x), h(x))=1$, hence by the Bezout identity:
+  $$\exists~ a(x), b(x) \in \mahtbb{F}_q[x] ~\text{s.th.}~ a(x) g(x) + b(x) h(x) = gcd(g, h)=1$$
+
+  Define $c(x) := a(x) g(x) = 1-b(x) h(x)$.
+
+  Clearly $c(x) \in C$. Furthermore, if $p(x) g(x) \in C$ (ie. $p(x)\cdot g(x)$ is any codeword in $C$), then
+  \begin{align*}
+    c(x) p(x) g(x) &= \underbrace{(1-b(x)h(x))}_{c(x)} p(x) g(x)\\
+                   &= p(x)g(x) - b(x)h(x)p(x)g(x)\\
+                   &(recall: g(x)h(x)=x^n-1)\\
+                   &= p(x) g(x) - b(x) p(x) \underbrace{g(x) h(x)}_{=x^n-1}\\
+                   &= p(x) g(x) \pmod{x^n -1}
+  \end{align*}
+
+  So $c(x)$ is an identity element for $C$, and hence is unique.
+\end{proof}
+
+Recall:
+\begin{itemize}
+  \item[nilpotent] $a \in A$ s.th. $a^n=0$ for $n>0$
+  \item[idempotent] $e \in A$ s.th. $e^2 = e$
+\end{itemize}
+
+\begin{defn}{6.4.2}[primitive idempotent]
+  the idempotent of an irreducible cyclic code $M_i^-$ is called a \emph{primitive idempotent}, denoted by $\theta_i(x)$.
+\end{defn}
+
+\begin{thm}{6.4.3}
+  if $C_1, C_2$ are cyclic codes with idempotents $c_1(x), c_2(x)$, then
+  \begin{enumerate}[i.]
+    \item $C_1 \cap C_2$ has idempotent $c_1(x) \cdot c_2(x)$
+    \item $C_1 + C_2$, ie. the set of all words $a+b$ with $a \in C_1,~ b \in C_2$, has idempotent $c_1(x) + c_2(x) + c_1(x) \cdot c_2(x) \in C_1+C_2$.
+  \end{enumerate}
+\end{thm}
+
+\begin{thm}{6.4.4}
+  for the primitive idempotent we have:
+  \begin{enumerate}[i.]
+    \item $\theta_i(x) \theta_j(x)=0$ if $i \neq j$
+    \item $\sum_{i=1}^t \theta_i(x) = 1$
+    \item $1+ \theta_{i_1}(x) + \theta_{i_2}(x) + \ldots + \theta_{i_r}(x)$ is the idempotent of the code with generator $f_{i_1}(x) \cdot f_{i_2}(x) \cdot \ldots \cdot f_{i_r}(x)$.
+  \end{enumerate}
+\end{thm}
+
+\subsection{Trace representation}
 
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