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+# Coding and Information Theory Crash Course
+
+## Encoding
+
+Let $A,B$ be two finite sets with size $a,b$ respectively.
+
+Let $S(A)=\bigcup_{r=1}^{\infty}A^r$ be the word semigroup generated by $A$.
+
+A one-to-one mapping $f:A\to S(B)$ is called a code with message alphabet $A$ and encoded alphabet $B$.
+
+Example:
+
+- $A=$ RGB color space
+- $B=\{0\sim 255\}$
+- $f:A\to B^n$ is a code
+
+For example, $f(white)=(255,255,255)$, $f(green)=(0,255,0)$
+
+### Uniquely decipherable codes
+
+A code $f:A\to S(B)$ is called uniquely decipherable if the extension code
+
+$$
+\tilde{f}:S(A)\to S(B)=f(a_1)f(a_2)\cdots f(a_n)
+$$
+
+is one-to-one.
+
+Example:
+
+- $A=\{a,b,c,d\}$
+- $B=\{0,1\}$
+- $f(a)=00$, $f(b)=01$, $f(c)=10$, $f(d)=11$
+
+is uniquely decipherable.
+
+- $f(a)=0$, $f(b)=1$, $f(c)=10$, $f(d)=11$
+
+is not uniquely decipherable.
+
+Since $\tilde{f}(ba)=10=\tilde{f}(c)$
+
+#### Irreducible codes
+
+A code $f:A\to S(B)$ is called irreducible if for any $x,y\in A$, $f(y)\neq f(x)w$ for some $w\in S(B)$.
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