Sunflowers of Reed--Solomon Codes
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≥3 and length ℓ≥2k−1, we give an explicit recursive construction with Ωk,ℓ(q⌊ℓ/(2k−1)⌋) petals and a greedy existence argument with Ωk,ℓ(qℓ−2k+2) petals as q→∞. We also apply the greedy argument to obtain families of [ℓ,k]q MDS codes of size Ωk,ℓ(q2(ℓ−2k+2)), whose pairwise intersections have dimension at most one but need not be equal.
Related
- Algorithmic List Decoding of Reed-Solomon Codes up to Capacity
- Constructions of LCPs and LCD codes from twisted Reed-Solomon codes
- Reed-Solomon Codes at Capacity: Algorithmic List-Decoding and Proximity Gaps
- New families of additive non-RS MDS codes and ACD codes from twisted Reed-Solomon codes
- Combinatorial limitations of list-decoding Reed-Solomon codes
BibTeX
@misc{con2026sunflowers,
title = {{Sunflowers of Reed--Solomon Codes}},
author = {Roni Con and Anina Gruica and Maria Montanucci and Ferdinando Zullo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33512},
eprint = {2609.33512},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33512},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sunflowers of Reed--Solomon Codes
AU - Roni Con
AU - Anina Gruica
AU - Maria Montanucci
AU - Ferdinando Zullo
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33512
UR - https://arxiv.org/abs/2609.33512
AB - 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≥3 and length ℓ≥2k−1, we give an explicit recursive construction with Ω_k,ℓ(q^⌊ℓ/(2k−1)⌋) petals and a greedy existence argument with Ω_k,ℓ(q^ℓ−2k+2) petals as q→∞. We also apply the greedy argument to obtain families of [ℓ,k]_q MDS codes of size Ω_k,ℓ(q^2(ℓ−2k+2)), whose pairwise intersections have dimension at most one but need not be equal.
ER -