Higher-dimensional product expansion conjecture for Reed–Solomon codes

Resolve the higher-dimensional conjecture formulated by Bafna and Vyas concerning product expansion for Reed–Solomon codes evaluated on multiplicative subgroups of coprime orders.

Background

The paper studies product expansion of tuples of Reed–Solomon codes evaluated on norm-one sets and proves a uniform product-expansion theorem for that setting. It contrasts this result with the work of Bafna and Vyas, who considered Reed–Solomon codes on multiplicative subgroups of coprime orders and formulated a higher-dimensional conjecture.

The authors explicitly state that their norm-one evaluation sets satisfy different arithmetic constraints and that their result does not resolve the Bafna–Vyas conjecture. Thus, establishing the conjectured higher-dimensional product-expansion statement in the coprime-order multiplicative-subgroup setting remains unresolved.

References

Bafna and Vyas studied product expansion for Reed-Solomon codes on multiplicative subgroups of coprime orders, proving a two-factor result and formulating a higher-dimensional conjecture. Our norm-one evaluation sets satisfy different arithmetic constraints, so our result is complementary and does not resolve their conjecture.

— Cubical Sheaf Complexes with Constant Expansion with Applications to Asymptotically Good qLTCs  (2609.28028 - Chen et al., 23 Sep 2026) in Section ‘Related work’

Their qLTC theorem assumes that certain Reed-Solomon code tuples have product expansion with a constant independent of their lengths, which is their Conjecture~5.6. In this work, we prove this expansion property for explicit norm-one families, using polynomial hulls and the inner-generation theorem proved in Appendix~\ref{sec:algebra}. In particular, our result (see Theorem~\ref{thm:rs-product-expansion}) covers the parameter family used in their construction, thereby partially resolving their conjecture.

— Cubical Sheaf Complexes with Constant Expansion with Applications to Asymptotically Good qLTCs  (2609.28028 - Chen et al., 23 Sep 2026) in Paragraph ‘Comparison with concurrent works’ in Section ‘Related work’