I need a check on the following exercise.
Let $C$ be the code over $F_2$ of length $7$, whose idempotent polynomial is $e(x)=1+x^3+x^5 + x^6$. Find its generator polynomial $g(x)$ and use the BCH bound to study the minimum distance of the code
By thoery I have that $\text{g.c.d }(x^7-1,e(x)) = g(x)$, where the division has to be in $F_2[x]$. By the euclidean algorithm, I found that the greatest common divisor is $g(x)=x^4+x^3+x^2 +1$, and so this is the generator polynomial.
To apply the BCH bound, I have to study the defining set $T$ which is $T = \cup_i C_i$ where $C_i$ is the $i$-th 2-cyclotomic coset modulo $7$ that comes from the generator polynomial $g$.
I have $C_0=\{0\}, C_1=\{1,2,4\} , C_2=\{ 3,5,6 \}$. I note that $g = (x+1)(x^3+x+1)$. The first factor corresponds to $C_0$, the second one to $C_1$. Therefore $T = \{0,1,2,4\}$ and hence, by the BCH bound, the minimum distance $d \geq 3$.
Is everything okay?
Of course, I could find the check polynomial, and hence the parity check matrix and study its columns in order to find the minimum distance.
But BCH bound says $d_{min}\geq N_0+1=4,$ where $N_0$ is the number of consecutive zeroes.