Non-Reed-Solomon Type MDS Codes from Elliptic Curves
Puyin Wang, Wei Liu, Jinquan Luo, Dengxin Zhai
TL;DR
This paper constructs new families of MDS codes from elliptic curves by leveraging the curve’s group structure and using divisors formed from a maximal subgroup, achieving parameters close to the bound $$(q + 1 + \lfloor 2\sqrt{q} \rfloor)/2$$ and yielding non-Reed–Solomon MDS codes. Equivalence to RS codes is disproved through an explicit analysis of Schur (Hadamard) products of generator matrices, demonstrating that the Schur-square dimensions differ from those of RS codes in the relevant ranges. The construction extends known parameter ranges for elliptic MDS codes and provides evidence supporting the tightness of upper bounds for algebraic-geometry MDS codes, while highlighting the need for new inequivalence techniques beyond Schur products. The results also illustrate how abelian group structures on elliptic curves can be exploited to realize longer MDS codes and enrich the landscape of non-RS MDS code constructions with practical implications for communication and storage systems.
Abstract
New families of maximum distance separable (MDS) codes are constructed from elliptic curves by exploiting their group structures. In contrast to classical constructions based on divisors supported at a single rational point, the proposed approach employs divisors formed by multiple distinct points constituting a maximal subgroup of the curve. The resulting codes achieve parameters approaching the theoretical upper bound $(q + 1 + \lfloor 2\sqrt{q} \rfloor)/2$ and include non Reed-Solomon (RS) MDS codes. The inequivalence of these codes to RS codes is established through an explicit analysis on the rank of the Schur product of their generator matrices. These results extend the known parameter range of elliptic MDS codes and provide additional evidence supporting the tightness of existing upper bounds for algebraic geometry MDS codes.