Elliptic curve codes yield new pairs of optimal error correcting codes
Linear Complementary Pairs of MDS Codes from Elliptic Curves
Information Theory
Summary
Error-correcting codes help computers and phones fix mistakes when sending information. The paper looks at special pairs of these codes, called Linear Complementary Pairs (LCPs), made from the mathematics of elliptic curves. The authors find conditions on these codes and show some limits about their construction. They also find ways to build new, very good code pairs by focusing on specific parts of the elliptic curve, creating families of codes that work well even for very long messages.
What this means in practice
- •For communications engineers: Design error-correcting code pairs with improved performance by using elliptic curve structures restricted to subgroup cosets for reliable data transmission.
- •For coding theory developers: Generate new infinite families of optimal error-correcting code pairs from specialized elliptic curves to expand available coding options for various system demands.
Authors
Adler Marques
Abstract
In this paper, we investigate the Linear Complementary Pairs (LCPs) of Maximum Distance Separable (MDS) codes that arise from elliptic curves. We establish geometric conditions for LCPs of elliptic codes and prove that they cannot consist entirely of MDS codes when evaluated over all rational points of the curve. To address this, we construct LCPs of MDS elliptic codes by restricting evaluation divisors to subgroup cosets. We apply this method to obtain explicit infinite families of LCPs of MDS codes over Mordell and supersingular elliptic curves, including large-length constructions over quadratic extensions.