Sunflowers of Reed--Solomon Codes

2026-09-27

AI Summary

This study investigates sunflower structures of Reed-Solomon codes in Grassmannian spaces, constructing code families whose pairwise intersections yield a fixed subspace. We introduce the first formal definition of Reed-Solomon code sunflowers and propose an algebraic criterion based on generalized Vandermonde matrices. By integrating extremal subspace combinatorics with affine equivalence analysis, explicit recursive constructions and greedy algorithms are developed for their resolution. The principal contributions include precisely determining the maximum family size in the two-dimensional case, providing an explicit construction containing Ω(q^⌊ℓ/(2k−1)⌋) petals for k ≥ 3, and establishing existence bounds for MDS code families. Overall, this work offers a novel framework for addressing extremal problems within algebraic coding theory.

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 $\Omega_{k,\ell}(q^{\lfloor\ell/(2k-1)\rfloor})$ petals and a greedy existence argument with $\Omega_{k,\ell}(q^{\ell-2k+2})$ petals as $q\to\infty$. We also apply the greedy argument to obtain families of $[\ell,k]_q$ MDS codes of size $\Omega_{k,\ell}(q^{2(\ell-2k+2)})$, whose pairwise intersections have dimension at most one but need not be equal.

Innovation

• Reed-Solomon sunflowers
• generalized V-matrices
• constant-dimension subspace codes
• MDS codes
• extremal subspace combinatorics