Conjecture on the parameters and weight distribution of codes from almost bent exponents

Let $n=2m+1$, let $e$ be one of the almost bent exponents listed in the source's table, and choose $t$ with $\gcd(t,n)=1$. Define

$$f(x)=(x^{2^t}+x)^e$$

and let $\mathcal{C}_f$ be the associated linear code.

Parameter and weight-distribution conjecture. The parameters of $\mathcal{C}_f$ should be the same as those of the code in the cited theorem, namely the theorem's parameters for the corresponding construction; moreover, the weight distribution of $\mathcal{C}_f$ should be determined.

This conjecture is motivated by experimental results and extends the observed code-parameter phenomenon to the listed almost bent exponents. The supplied text does not include the referenced table or theorem, so the exact parameter tuple and the claimed weight distribution remain to be determined.