I will begin with the definition of BCH code.
Definition: A cyclic code of length $\tilde{n}$ over $\mathbb{F}_q$ is called a BCH code of designed distance $d$ if its generator polynomial $g(x)$ is the least common multiple of the minimal polynomials of $\alpha^l, \alpha^{l+1},...,\alpha^{l+d-2}$ for some $l$, where $\alpha$ is a primitive $n^{th}$ root of unity. If $l=1$, then the code is called narrow-sense BCH code. If $\tilde{n}= q^{\tilde{m}}-1$, i.e. $\alpha$ is the primitive element of $\mathbb{F}_{q^{\tilde{m}}}$, where $\tilde{m}$ being an integer, then the BCH code is called primitive.
Now, a linear code is symmetric if and only if the all one code vector $\bf{1}_{\tilde{n}}$ is a valid code vector. I know that if $(x+1)\nmid g(x)$ (where $g(x)$ is the generator polynomial of a BCH code) then the code will be symmetric. With the aforementioned definition of BCH codes, for a binary narrow-sense primitive BCH code, I can prove that $(x+1)\nmid g(x)$; but I don't understand how this condition implies that the code will be symmetric?
I am writing a journal paper where I have state this fact that a binary narrow-sense primitive BCH code is always symmetric. But giving the proof of this statement would be beyond the scope of my research paper. Therefore, I am looking for a book or a research paper which has the proof of this statement, so that I can cite their result. So far I am not able to find any book which has the proof of that statement. I need your help in this regard.