  Let $\mathbb{F}_q[t]$ denote the ring of polynomials over $\mathbb{F}_q$, the
finite field of $q$ elements. Suppose the characteristic of $\mathbb{F}_q$ is
not $2$ or $3$. In this paper, we prove an $\mathbb{F}_q[t]$-analogue of
results related to the conjecture of Erd\H{o}s on the existence of infinite
Sidon sequence of positive integers which is an asymptotic basis of order 3. We
prove that there exists a $B_2[2]$ sequence of non-zero polynomials in
$\mathbb{F}_q[t]$, which is an asymptotic basis of order $3$. We also prove
that for any $\varepsilon> 0$, there exists a sequence of non-zero polynomials
in $\mathbb{F}_q[t]$, which is a Sidon basis of order $3 + \varepsilon$. In
other words, there exists a sequence of non-zero polynomials in
$\mathbb{F}_q[t]$ such that any $n \in \mathbb{F}_q[t]$ of sufficiently large
degree can be expressed as a sum of four elements of the sequence, where one of
them has a degree less than or equal to $\varepsilon \text{deg } n.$
