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

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

@@ -2070,6 +2070,29 @@
   since $C$ is an ideal of $R_n$ and $g(x) \in C$ and $q(x) \in R_n$.
 \end{proof}
 
+\begin{thm}{12.15}
+  $C$ cyclic $[n,k]$-code with check polynomial
+  $$h(x) = h_0 + h_1 x + \ldots + h_k x^k$$
+
+  Then
+  \begin{enumerate}[i.]
+    \item a PCM for $C$ is
+      $$H =
+      \begin{bmatrix}
+        h_k & h_{k-1} & h_{k-2} & \cdots & h_0 & 0 & 0 & \cdots & 0\\
+        0 & h_k & h_{k-1} & h_{k-2} & \cdots & h_0 & 0 & \cdots & 0\\
+        0 & 0 & h_k & h_{k-1} & h_{k-2} & \cdots & h_0 & \cdots & 0\\
+          & \vdots & & & & & & \vdots\\
+        0 & 0 & \cdots & 0 & h_k & h_{k-1} & h_{k-2} & \cdots & h_0\\
+      \end{bmatrix}
+      $$
+
+    \item $C^\perp$ is a cyclic code generated by
+      $$\bar{h}(x)=h_k + h_{k-1} x + \ldots + h_0 x^k$$
+  \end{enumerate}
+\end{thm}
+
+\vspace{1.0cm}
 \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.
@@ -2078,22 +2101,38 @@ \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}.
+
+  \vspace{0.6cm}
+  ie.
+  $$M_i^+= \langle f_i(x) \rangle, ~~~~ M_i^-= \big\langle \frac{(x^n-1)}{f_i(x)} \big\rangle$$
 \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 $\beta_i$.
+  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 $E \supset \mathbb{F}_q$,\\
+  \hspace*{2em}ie. $f_i(\beta_i)=0$, with $f_i(x)$ 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:
+\vspace{0.5cm}
+
+Specify a cyclic code by requiring all codewords to have certain zeros, ie.\\
+let the set $\alpha_1, \alpha_2, \ldots, \alpha_s$; define the code $C$ by
+$$c(x) \in C ~\text{iff}~ c(\alpha_i)=0 ~\forall i \in [s]$$
+then $C$ cyclic, and the generator polynomial is the lcm of the minimal polynomials of $\alpha_1, \alpha_2, \ldots, \alpha_s$.
+
+
+\vspace{0.6cm}
+Take the $sm \times n$ matrix with columns $1, \alpha_i, \alpha_i^2, \ldots,
+\alpha_i^{n-1}$, denoted $H$, with entries in $\mathbb{F}_q$. Then
 
-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 $\alpha_1, \alpha_2, \ldots, \alpha_s$.
+\begin{align*}
+  cH^T&=0, ~\text{with}~ c=(c_0, c_1, \ldots, c_{n-1})\\
+  c(\alpha_i)&=0 ~\forall i \in [s], ~\text{with}~ c(x) \in C
+\end{align*}
+
 
 \subsection{Hamming cyclic codes}
 \begin{thm}{6.3.1}
@@ -2275,7 +2314,7 @@
   $\Phi$ is an isomorphism of the ring $(T, +, \circ)$ onto the ring $(T, +, *)$.
 \end{remark}
 
-\begin{lemma}{6.5.4}
+\begin{lemma}{6.5.4} \label{6.5.4}
   let $V$ a cyclic code over $\mathbb{F}_q$ generated by
   $$g(x) = \prod_{k \in K} (x - \beta^k)$$
 
@@ -2289,12 +2328,34 @@
   Thus at the equation $A(X) := \sum_{j=1}^n a(\beta^j) X^{n-j}$, it will be $0$  between $1 \leq j \leq d-1$, so that degree is $n-d$.
 \end{proof}
 
-\begin{thm}{6.5.5}
+\begin{thm}{6.5.5} \label{6.5.5}
   if there are $r$ n-th roots of unity which are zeros of the Mattson-Solomon polynomial $A$ of a word $a$, then $w(a) = n-r$.
 \end{thm}
 \begin{proof}
   consequence of Lemma \ref{6.5.3}.
 \end{proof}
+
+
+
+\vspace{1.0cm}
+\section{BCH Codes}
+
+\begin{defn}{6.6.1}[BCH Code]
+  a cyclic code of length $n$ over $\mathbb{F}_q$ is called \emph{BCH code of designed distance $\delta$} if its generator $g(x)$ is the lcm of the minimal polynomials of $\beta^l, \beta^{l+1}, \ldots, \beta^{l+\delta-2}$ for some $l$, where $\beta$ is a primitive n-th root of unity.
+\end{defn}
+
+Usually $l=1$; if $n=q^m-1$ (ie. $\beta$ a primitive element of $\mathbb{F}_q^m$), then the BCH code is called \emph{primitive}.
+
+
+\begin{thm}{6.6.2}[BCH Bound]
+  the minimum distnce of a BCH code with designed distance $d$ is $\geq d$.
+\end{thm}
+\begin{proof}
+  wlog. take $l=1$. By Lemma \ref{6.5.4}, the degree of the Mattson-Solomon polynomial of a codeword $c$ is $\leq n-d$.
+
+  Therefore, in theorem \ref{6.5.5} we have $r \leq n-d$, ie. $w(c) \geq d$.
+\end{proof}
+
 
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