Section 1
BCH Codes
Show that $x^3+x+1$ is irreducible over $G F_2$.
Let $\alpha$ be a root of $x^3+x+1$ in $G F_8$. Construct the minimal polynomial of $\alpha^3$.
Use the polynomial $x^3+x+1$ to construct a $\mathrm{BCH}$ code that corrects a single error. List all of the plaintext and codeword polynomials and observe that the minimum distance between codewords is three.
Show that $x^2+1$ is irreducible over $G F_3$.
Let $\alpha$ be a root of $x^2+1$ in $G F_9$. Construct the minimal polynomial of $\alpha+1$.
Show that $x^4+x^3+1$ is irreducible over $G F_2$ by showing that if $\alpha$ is a root of $x^4+x^3+1$, then the order of $\alpha$ is 15 .
Let $\theta$ be a root of $x^4+x^3+1$ in $G F_{16}$. Construct the minimal polynomial of $\theta^3+\theta^2$.
Show that $x^5+x^3+x+1$ is irreducible over $G F_3$ by showing that if $\theta$ is a root of $x^5+x^3+x+1$, then the smallest field containing $\theta$ has at least $3^5=243$ elements.
Let $\theta$ be a root of $x^5+x^3+x+1$ in the field $G F_{3^5}$. Construct the minimal polynomial of $\theta^4+2 \theta$.
Show that $x^2-2$ is irreducible over $G F_5$.
Let $\theta$ be a root of $x^2-2$ in the field $G F_{5^2}=G F_{25}$. Construct the minimal polynomial of $3 \theta+2$.
From the equation $(a+b)^p=a^p+b^p$ in $G F_{p^n}$ (see Theorem 10.5), derive the equation $\left(a_1+a_2+\cdots+a_k\right)^p=a_1^p+a_2^p+\cdots+a_k^p$ by induction on $k$.
From the equation $(a+b)^p=a^p+b^p$ in $G F_{p^n}$, derive the equation $(a-b)^p=$ $a^p-b^p$.
The code that encodes a word of length 5 by repeating it three times is a $(15,5)$ polynomial code. What is its generator polynomial?
The triple-repetition code (see page 102) is not a polynomial code but can be described as encoding the polynomial $p(x)$ by the polynomial $\left(1+x+x^2\right) p\left(x^3\right)$. Verify both parts of this statement.