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

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

@@ -107,7 +107,7 @@
 \maketitle
 
 \begin{abstract}
-    Notes taken while studying Algebraic Coding Theory, mostly from Hill book \cite{hill} and Lint book \cite{lint}.
+    Notes taken while studying Algebraic Coding Theory, mostly from Hill's book \cite{hill} and Lint's book \cite{lint}.
 
     Usually while reading books and papers I take handwritten notes in a notebook, this document contains some of them re-written to $LaTeX$.
 
@@ -2347,7 +2347,7 @@
 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]
+\begin{thm}{6.6.2}[BCH Bound] \label{6.6.2}
   the minimum distance of a BCH code with designed distance $d$ is $\geq d$.
 \end{thm}
 \begin{proof}
@@ -2392,6 +2392,135 @@ \end{cor}
 \begin{cor}{6.6.16}
   the minimum weight of a primitive binary BCH code is odd.
 \end{cor}
+
+
+
+
+\section{Reed-Solomon Codes}
+
+\begin{defn}{6.8.1}[Reed-Solomon code]
+  a \emph{Reed-Solomon code} (RS code) is a primitive BCH code of length $n=q-1$ over $\mathbb{F}_q$.
+
+  With generator
+  $$g(x) = \prod_{i=1}^{d-1} (x - \alpha^i)$$
+  where $\alpha$ is primitive in $\mathbb{F}_q$.
+\end{defn}
+
+\begin{remark}{}observe:\\
+  \begin{itemize}
+    \item by \ref{6.6.2}, minimum distance $\geq d$
+    \item by 6.2, has dimension $k=n-d+1$
+  \end{itemize}
+  Therefore, Corollary 5.2.2 implies that minimum distance is $d$ and the RS code is a MDS.
+
+  This is useful for when errors are in bursts (ie. several errors together).
+\end{remark}
+
+\vspace{0.6cm}
+Let $n=q-1$, $\alpha$ a primitive element of $\mathbb{F}_q$.
+
+$$a=(a_0, a_1, \ldots, a_{k-1}) \in \mathbb{F}_q^k \Longleftrightarrow a(x) = a_0 + a_1 x + \ldots + a_{k-1}x^{k-1}$$
+
+Then
+$$C= \{ (c_0, c_1, \ldots, c_{n-1}) ~|~ c_i = a(\alpha^i),~ 0 \leq i <n,~ a \in \mathbb{F}_q^k \}$$
+is the RS code with $d = n - k +1$.
+
+To see it: $C$ is cyclic. By definition of $C$ and Lemma \ref{6.5.3}, implies that a codeword $c$ has $n \cdot a(x)$ as its Mattson-Solomon polynomial.
+
+Since $deg(a(x)) \leq k-1$,
+$$c(\alpha^i) = 0 ~~\forall i \in [1, n-k]$$
+
+Extend the words of $C$ by adjoining a symbol $c_n = a(0)$, then
+$$\sum_{i=0}^n c_i = a_0 q = 0$$
+$\Longrightarrow~$ we obtain $\overline{C}$.
+
+
+If $c_n=0$, ie. $c(1)=0$, then the word has weight $\geq n-k+2$. Also true if $c_n \neq 0$.
+$\Longrightarrow~$ so $\overline{C}$ is also a MDS code.
+
+
+\begin{defn}{6.8.2}
+  the generalized R-S code, $GRS_k(a, v)$, has codewords all
+  $$(v_1 f(\alpha_1), v_2 f(\alpha_2), \ldots, v_n f(\alpha_n))$$
+  where $f$ runs through the set of polynomials of degree $<k$ in $\mathbb{F}_{q^m}[X]$, ie. $f \in \mathbb{F}_{q^m}[X]^{<k}$.
+\end{defn}
+
+A GRS and its dual are MDS codes.
+
+\begin{thm}{6.8.3}
+  the dual of an MDS code is also an MDS code.
+\end{thm}
+\begin{proof}
+  let $G=(I_k~ P)$ be GM of $C$.
+
+  Since $C$ has min. weight $d$, every set of $d-1=n-k$ columns of the PCM $H:=(-P^T ~ I_{n-k})$ is linearly independent.
+
+  Hence every square submatrix of $H$ is nonsingular, ie. no codeword of $C^{\perp}$ has $n-k$ zeros.
+
+  So $C^{\perp}$ is an $[n, n-k, k+1]$ code, ie. MDS.
+\end{proof}
+
+
+
+\vspace{1.0cm}
+\subsection{Example of Hamming code (cyclic) and of Reed-Solomon code (cyclic and MDS)}
+
+\paragraph{(7,4)-Hamming code over $\mathbb{F}_2$}
+
+$$(n, k) \Longrightarrow~ n=7 (\text{length}), k=4 (\text{message length})$$
+
+Generator polynomial: $g(x) = x^3 + x +1$.
+
+Parity bits: $r=3$
+
+Message: $m(x) \in \mathbb{F}_2[X]^{\leq 3}$, encoded as $c(x) = m(x) g(x) \pmod{(x^7-1)}$
+
+eg: for $m(x) = x^3 + x + 1$,
+$$c(x) = (x^3 + x +1) \cdot (x^3 +x+1) \pmod({x^7-1}) = x^6 + x^2 + 1$$
+$$\Longrightarrow~ c=(1000101)$$
+
+Hamming code is cyclic:\\
+when $c(x) \in C$, then $x \cdot c(x) \pmod({x^n-1}) \in C$,
+
+because $C=\langle g(x) \rangle$, ie. all codewords are multiples of $g(x)$,
+
+so for $c(x) = m(x) \cdot g(x)$, then
+$$x \cdot c(x) = x \cdot m(x) \cdot g(x)$$
+is still a multiple of $g(x)$, hence a valid codeword.
+
+
+\paragraph{(6,4)-Reed-Solomon code over $\mathbb{F}_7$}
+
+$$n=6,~ k=4,~ \text{minimum distance:}~ d=n-k+1 = 6-4+1 = 3$$
+
+Choose a primitive element, $\alpha=3 \in \mathbb{F}_7$.
+
+Generator polynomial:
+\begin{align*}
+  g(x) &= (x-\alpha) \cdot (x - \alpha^2)\\
+       &= x^2 -2x -3x +6\\
+       &= x^2 -5x +6 = x^2 +2x +6 \pmod{7}
+\end{align*}
+
+message: $m(x) = 1 + 2x + x^2 + x^3$
+
+Then,
+\begin{itemize}
+  \item[cyclic form:] $c(x) = m(x) g(x) \pmod{(x^6-1)}$\\
+    $c(x) \in C$ iff $c(\alpha^i)=0 ~\forall~ i \in [n-k]$
+\item[evaluation form:] $c = (m(1), m(\alpha), m(\alpha^2), \ldots, m(\alpha^6))$
+\end{itemize}
+
+RS code is cyclic. And it is MDS:
+\begin{itemize}
+  \item a polynomial of degree $<k$ is uniquely determined by $k$ points
+  \item if two polynomials agree in $k$ positions, are the same polynomial\\
+    $\Longrightarrow$ maximum number of equal positions between distinct codewords $= k-1$
+\end{itemize}
+
+Therefore, $d=n-(k-1) = n-k+1$, ie. Singleton bound (with equality!).
+
+
 
 
 \bibliographystyle{unsrt}