3-Designs from $\mathrm{GL}_2(\mathbb{F}_q)$-Invariant Subspaces of $\mathbb F_q[X,Y]_k$

We present a uniform framework for constructing $3$-designs from $\mathrm{GL}_2(\mathbb F_q)$-invariant subspaces of $\mathbb F_q[X,Y]_k$, the space of homogeneous polynomials of degree $k$. Given such a subspace $W$, we associate a $\mathrm{PGL}_2(\mathbb F_q)$-invariant family of $k$-subsets of $\mathbb P^1(\mathbb F_q)$. Whenever this family is nonempty, it forms a $3\text{-}(q+1,k,\lambda)$ design. When $k\le q$, the evaluation map on $\mathbb P^1(\mathbb F_q)$ identifies $W$ with a subcode $C_W$ of the projective Reed--Solomon code. We also show that the supports of minimum-weight codewords in $C_W$, as well as the supports of suitable fixed-weight codewords in the dual code $C_W^\perp$, yield further $3$-designs. Via the Cayley transform, the construction is transferred to the unit circle $U_{q+1}\subseteq \mathbb F_{q^2}^{\times}$, where the block conditions become explicit linear relations among elementary symmetric polynomials. Applying this framework to the Lucas subspaces, we obtain explicit block descriptions, classify the cases in which the defining conditions reduce to a single equation, and establish several emptiness and nonemptiness results. In particular, for $q=p^e$ and $k=p^m+1$, we show that the associated block family is nonempty if and only if $m\mid e$, in which case it yields the Steiner system $S(3,p^m+1,q+1)$. Finally, in the ternary case $p=3$ and $k=7$, we use the weight distribution of the ternary Melas code to determine the design parameters left undetermined by Xu et al. (Designs, Codes and Cryptography: Vol. 92, 2024).