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

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

@@ -2143,7 +2143,7 @@
   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:
-  $$\exists~ a(x), b(x) \in \mahtbb{F}_q[x] ~\text{s.th.}~ a(x) g(x) + b(x) h(x) = gcd(g, h)=1$$
+  $$\exists~ a(x), b(x) \in \mattbb{F}_q[x] ~\text{s.th.}~ a(x) g(x) + b(x) h(x) = gcd(g, h)=1$$
 
   Define $c(x) := a(x) g(x) = 1-b(x) h(x)$.
 
@@ -2236,6 +2236,62 @@ So we observer $c(x) h(x) = 0$, from Theorem \ref{12.14} this means that $h(x)$ is the check polynomial,\\
   ie. we have a parity check equation for the code $V$.
 
   Since $h(x)$ irreducible, $x^k \cdot h(x^{-1})$ is the check polynomial for $V$, therefore, $V$ is an irreducible cyclic $[n, k]$-code.
+\end{proof}
+
+\subsection{Mattson-Solomon Polynomial}
+
+\begin{defn}{}[Mattson-Solomon polynomial]
+Let $\beta$ primitive n-th root of unity in the extension field $E$ of $\mathbb{F}_q$.
+
+Let $T$ the set of polynomials over $E$ with degree at most $n-1$, ie.
+$$T = \{ p(x) \in E ~|~ deg(p(x)) \leq n-1 \}$$
+
+Define $\Phi: T \longrightarrow T$ as:
+
+let $a(x) \in T$, then $A(X) = (\Phi_a)(X)$ is defined by
+$$A(X) := \sum_{j=1}^n a(\beta^j) X^{n-j}$$
+
+If $a=(a_0, a_1, \ldots, a_{n-1})$, the polynomial $A(X)$ obtained from $a_0 + a_1 x + \lldots a_{n-1} x^{n-1}$ is called the \emph{Mattson-Solomon polynomial} of the vector $a$.
+\end{defn}
+
+\begin{lemma}{6.5.3} \label{6.5.3}
+  the inverse of $\Phi$ is given by
+  $$a(x) = n^{-1}(\Phi A)(x^{-1}) \pmod{x^n -1}$$
+\end{lemma}
+\begin{proof}
+  $$A(\beta^k) = \sum_{j=1}^n \sum_{i=0}^{n-1} a_i \beta^{ij} \beta^{-kj} = \sum_{i=0}^{n-1} a_i \sum_{j=1}^n \beta^{(i-k)j}=n a_k$$
+\end{proof}
+
+\begin{remark}{}
+  Notation:
+  \begin{itemize}
+    \item[] $\circ$: multiplication of polynomials $mod (x^n -1)$
+    \item[] $*$: defined by
+      $$(\sum a_i x^i) * (\sum b_i x^i) := \sum a_i b_i x^i$$
+  \end{itemize}
+
+  $\Phi$ is an isomorphism of the ring $(T, +, \circ)$ onto the ring $(T, +, *)$.
+\end{remark}
+
+\begin{lemma}{6.5.4}
+  let $V$ a cyclic code over $\mathbb{F}_q$ generated by
+  $$g(x) = \prod_{k \in K} (x - \beta^k)$$
+
+  Suppose $\{1, 2, \ldots, d-1 \} \subset K$ and $a \in V$.
+
+  Then the degree of the Mattson-Solomon polynomial $A$ of $a$ is at most $n-d$.
+\end{lemma}
+\begin{proof}
+  $a(\beta^j) = 0$ for $j \in [d]$ since $g(x)|a(x)$.
+
+  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}
+  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}
 
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