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

Abstract

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 $3\text{-}(q+1,k,λ)$ design. Via the Cayley transform, the construction is reformulated on the unit circle $U_{q+1}\subseteq \mathbb F_{q^2}^{\times}$, where the block conditions become explicit linear relations among elementary symmetric polynomials. This reformulation unifies several previously disparate constructions and simplifies a number of delicate ad hoc computations. 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 show that the associated block family is nonempty if and only if $d(C_W)=q+1-k$. Under this condition, 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. Applying this framework to the Lucas subspaces, which form a distinguished family of invariant 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.