Improved Field Size Bounds for Higher Order MDS Codes

• Published in: IEEE transactions on information theory Vol. 70; no. 10; pp. 6950 - 6960
• Main Authors: Brakensiek, Joshua, Dhar, Manik, Gopi, Sivakanth
• Format: Journal Article
• Language: English
• Published: IEEE 01.10.2024
• Subjects: Codes; Decoding; Generators; Linear codes; list-decoding; lower bounds; MDS codes; Parity check codes; Reed-Solomon codes; Tensors; Upper bound; upper bounds
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2024.3449030

Higher order MDS codes are an interesting generalization of MDS codes recently introduced by Brakensiek et al., (2023). 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 <inline-formula> <tex-math notation="LaTeX">\rm {MDS}(\ell) </tex-math></inline-formula> where <inline-formula> <tex-math notation="LaTeX">\ell </tex-math></inline-formula> denotes the order of generality, <inline-formula> <tex-math notation="LaTeX">\rm {MDS}(2) </tex-math></inline-formula> codes are equivalent to the usual MDS codes. The best prior lower bound on the field size of an <inline-formula> <tex-math notation="LaTeX">{[}n,k{]} </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">\rm {MDS}(\ell) </tex-math></inline-formula> codes is <inline-formula> <tex-math notation="LaTeX">\Omega _{\ell } (n^{\ell -1}) </tex-math></inline-formula>, whereas the best known (non-explicit) upper bound is <inline-formula> <tex-math notation="LaTeX">O_{\ell } (n^{k(\ell -1)}) </tex-math></inline-formula> which is exponential in the dimension. In this work, we nearly close this exponential gap between upper and lower bounds. We show that an <inline-formula> <tex-math notation="LaTeX">{[}n,k{]} </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">\rm {MDS}(3) </tex-math></inline-formula> codes requires a field of size <inline-formula> <tex-math notation="LaTeX">\Omega _{k}(n^{k-1}) </tex-math></inline-formula>, 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 by Shangguan and Tamo, (2020). We also give explicit constructions of <inline-formula> <tex-math notation="LaTeX">{[}n,k{]} </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">\rm {MDS}(\ell) </tex-math></inline-formula> code over fields of size <inline-formula> <tex-math notation="LaTeX">n^{(\ell k)^{O(\ell k)}} </tex-math></inline-formula>. The smallest non-trivial case where we still do not have optimal constructions is <inline-formula> <tex-math notation="LaTeX">{[}n,3{]} </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">\rm {MDS}(3) </tex-math></inline-formula>. In this case, the known lower bound on the field size is <inline-formula> <tex-math notation="LaTeX">\Omega (n^{2}) </tex-math></inline-formula> and the best known upper bounds are <inline-formula> <tex-math notation="LaTeX">O(n^{5}) </tex-math></inline-formula> for a non-explicit construction and <inline-formula> <tex-math notation="LaTeX">O(n^{32}) </tex-math></inline-formula> for an explicit construction. In this paper, we give an explicit construction over fields of size <inline-formula> <tex-math notation="LaTeX">O(n^{3}) </tex-math></inline-formula> which comes very close to being optimal.