Sunflowers of Reed--Solomon Codes

Source abstract

We introduce and study Reed--Solomon sunflowers, namely families of Reed--Solomon codes whose pairwise intersections are all equal to the same fixed subspace. This notion lies at the intersection of extremal subspace combinatorics and coding theory: it can be viewed as a structured version of the sunflower problem in the Grassmannian, and it naturally produces constant-dimension subspace codes with prescribed minimum distance. We focus mainly on the case in which the center is the one-dimensional space generated by the all-one vector. We give an algebraic criterion, expressed in terms of generalized $V$-matrices, ensuring that a family of Reed--Solomon codes forms such a sunflower. We then study the size of these families through counting and constructions. In dimension two, we show that all distinct Reed--Solomon codes form a sunflower and determine its size by counting Reed--Solomon codes up to affine equivalence of their evaluation vectors. For fixed dimension $k\geq3$ and length $\ell\geq2k-1$, we give an explicit recursive construction with $Ω_{k,\ell}(q^{\lfloor\ell/(2k-1)\rfloor})$ petals and a greedy existence argument with $Ω_{k,\ell}(q^{\ell-2k+2})$ petals as $q\to\infty$. We also apply the greedy argument to obtain families of $[\ell,k]_q$ MDS codes of size $Ω_{k,\ell}(q^{2(\ell-2k+2)})$, whose pairwise intersections have dimension at most one but need not be equal.