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.

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