On a class of twisted elliptic curve codes
Xiaofeng Liu, Jun Zhang, Fang-Wei Fu
TL;DR
This work introduces TECCs, a twisted extension of elliptic curve codes with a single twist, and develops explicit parity-check constructions using Weil differentials and the Riemann-Hurwitz formula. It derives self-duality criteria, determines minimum distances, and provides MDS, AMDS, self-dual, and MDS self-dual examples, including explicit parity-check matrices. It then studies the Schur-square dimensions to prove TECCs are not monomially equivalent to GRS or ECC codes in broad parameter ranges, highlighting TECCs as a distinct code family for cryptography and coding theory. The results pave the way for further twisted AG codes and decoding/cryptographic applications.
Abstract
Motivated by the studies of twisted generalized Reed-Solomon (TGRS) codes, we initiate the study of twisted elliptic curve codes (TECCs) in this paper. In particular, we study a class of TECCs with one twist. The parity-check matrices of the TECCs are explicitly given by computing the Weil differentials. Then the sufficient and necessary conditions of self-duality are presented. The minimum distances of the TECCs are also determined. Moreover, examples of MDS, AMDS, self-dual and MDS self-dual TECCs are given. Finally, we calculate the dimensions of the Schur squares of TECCs and show the non-equivalence between TECCs and ECCs/GRS codes.