Let $\mathcal{C}:=[n, k, d]_2$ denotes a binary linear code of block length $n$, dimension $k$, and minimum distance $d$. Now, for a positive integer $m$, a $[2^m-1, m+1, 2^{m-1}-1]_2$-BCH code has the following weight distribution:
$$ a_0 = 1 ,\\ a_{2^m - 1} = 1,\\ a_{2^{m-1}-1} = 2^{m}-1 ,\\ a_{2^{m-1}} = 2^m - 1,\\ a_{j} = 0,\ \mbox{for other j's}, $$ where $a_i$ denotes the number of codewords of weight $i$.
Now, I know that the above weight distribution is true, but I don't know the proof of it. Even if I knew the proof of this weight distribution, it would not help me because I am writing a research paper where I need to mention this weight distribution but giving the proof of it would go beyond the scope of my paper. Therefore I am searching for a published research paper or a book where I can find a theorem regarding the proof of weight distribution of the $[2^m-1, m+1, 2^{m-1}-1]_2$-BCH codes. If I find such a theorem, I would cite it straight away in my paper. Can somebody give the direction on where I might find the weight distribution of such BCH codes?
To clarify, the m-sequence code Dilip Sarwate mentioned in the comment has weight distribution:
$$ a_0 = 1 ,\\ a_{2^{m-1}} = 2^m - 1,\\ a_{j} = 0,\ \mbox{for other j's}, $$ since its nonzero codewords are precisely the cyclic shifts of the m-sequence.
Your code is this code, plus this code translated by adding the all 1 vector.
This gives you a full weight codeword as well as the complements of the m-sequence shifts, each of weight $2^{m-1}-1.$