How can I construct a generator polynomial for a BCH code $(7,3)$ code
over $GF(2^3)$ with designed distance $\delta =5$.
Observe that $$x^7-1 = (x+1)(x^3+x+1)(x^3+x^2+1)=m_0(x)m_1(x)m_2(x)$$ where $m_i(x)$ are the minimal polynomials for $\alpha^i$ where $\alpha$ is the primitive element of $$GF(2^3)= GF(2)/(y^3+y+1).$$ Also, $\alpha$ is the $7$-th root of unity and $m_1(x)=m_2(x)=m_4(x)$ and $m_3=m_5=m_6(x)$.
Note that the cyclotomic sets are $$C_0=\{0\}, \quad C_1= \{1,2,4\}, C_2= \{3,5,6\}.$$ Since the designed distance is $5$, I cant pick $\delta-1=4$ distinct consecutive elements from $$\alpha^a, \alpha^{a+1}, \cdots, \alpha^{a+\delta-2}$$ for any value of $a$ without having an element in the three cosets and that will make the degree of my polynomial to be $6$ or $7$ and I also want the weight of the generator to be at least $5$. The most natural choice would be $$1+x+x^2+x^3+x^4$$ but this can not work because it does not divide $x^7-1$. Also, it is not the lcm of any of the minimal polynomial. Please, How can i construct this generator polynomial. Any help will be appreciated.
Best Answer
The first thing you have to do is to create a Field with 8 elements and find a primite element of it. For this, it's sufficient to consider $\mathbb{F}_8 := \mathbb{F}_2/(x^3+x+1)$. Denote $\alpha := [x]$, then $\alpha^3=\alpha+1$. Finding a primitive element of the field means find a generator of the multiplicative group $(\mathbb{F}_8)^* := \mathbb{F}_8$. Note that $\vert (\mathbb{F}_8)^* \vert = 7$ and since every element $ 1 \neq \beta \in (\mathbb{F}_8)^*$ has a period that divide 7, each non trivial element of $(\mathbb{F}_8)^*$ is primitive. In conclusion we can take $\alpha$ as a primitive element of the field.
Since your code is a Reed Solomon code, the cyclotomic sets are made up by just one element, and you can consider the Defining set $\{ 1,2,3,4 \}$, which is also a complete defining set for a suitable polynomial $$g:=(x-\alpha)\cdot (x-\alpha^2) \cdot (x-\alpha^3) \cdot (x-\alpha^4) \in \mathbb{F}_8[x]$$ Computing $g$ we obtain $g = x^4+(\alpha+1)x^3+x^2+\alpha x + \alpha + 1$
Observe that this polynomial satisfies your request