Improved Field Size Bounds for Higher Order MDS Codes
Joshua Brakensiek, Manik Dhar, Sivakanth Gopi
TL;DR
This work substantially narrows the exponential gap for field sizes required by higher order MDS codes, establishing a near-tight lower bound |f| inom{n-2}{k-1}-1 for [n,k]-MDS(3) codes and translating this into MR tensor codes and LD-MDS implications. It introduces an explicit, general construction over fields of size n^{( k)^{O( k)}} for [n,k]-MDS(), and then provides concrete, near-optimal RS-based constructions: [n,3]-MDS(3) in O(n^3), [n,4]-MDS(3) in O(n^7), and [n,5]-MDS(3) in O(n^{50}), with duals yielding corresponding [n,n-k]-MDS(3) codes for k=3,4,5. The paper also develops determinant-based criteria for MDS( ll) in Reed-Solomon codes, connects to GM-MDS theory, and situates the results relative to prior doubly-exponential and probabilistic constructions, highlighting both the progress and remaining gaps toward optimal constructions in the general MDS( ll) regime. Overall, the results advance explicit code design for distributed storage and list-decoding applications by delivering stronger lower bounds and practical, scalable constructions. The work thus enhances understanding of field-size requirements for high-order MDS structures and informs future efforts toward truly optimal explicit codes.
Abstract
Higher order MDS codes are an interesting generalization of MDS codes recently introduced by Brakensiek, Gopi and Makam (IEEE Trans. Inf. Theory 2022). In later works, they were shown to be intimately connected to optimally list-decodable codes and maximally recoverable tensor codes. Therefore (explicit) constructions of higher order MDS codes over small fields is an important open problem. Higher order MDS codes are denoted by $\operatorname{MDS}(\ell)$ where $\ell$ denotes the order of generality, $\operatorname{MDS}(2)$ codes are equivalent to the usual MDS codes. The best prior lower bound on the field size of an $(n,k)$-$\operatorname{MDS}(\ell)$ codes is $Ω_\ell(n^{\ell-1})$, whereas the best known (non-explicit) upper bound is $O_\ell(n^{k(\ell-1)})$ which is exponential in the dimension. In this work, we nearly close this exponential gap between upper and lower bounds. We show that an $(n,k)$-$\operatorname{MDS}(3)$ codes requires a field of size $Ω_k(n^{k-1})$, which is close to the known upper bound. Using the connection between higher order MDS codes and optimally list-decodable codes, we show that even for a list size of 2, a code which meets the optimal list-decoding Singleton bound requires exponential field size; this resolves an open question from Shangguan and Tamo (STOC 2020 / SIAM J. on Computing 2023). We also give explicit constructions of $(n,k)$-$\operatorname{MDS}(\ell)$ code over fields of size $n^{(\ell k)^{O(\ell k)}}$. The smallest non-trivial case where we still do not have optimal constructions is $(n,3)$-$\operatorname{MDS}(3)$. In this case, the known lower bound on the field size is $Ω(n^2)$ and the best known upper bounds are $O(n^5)$ for a non-explicit construction and $O(n^{32})$ for an explicit construction. In this paper, we give an explicit construction over fields of size $O(n^3)$ which comes very close to being optimal.