On the MacWilliams Theorem over Codes and Lattices
Cryptography and Security
2024-07-31 v1
Abstract
Analogies between codes and lattices have been extensively studied for the last decades, in this dictionary, the MacWilliams identity is the finite analog of the Jacobi-Poisson formula of the Theta function. Motivated by the random theory of lattices, the statistical significance of MacWilliams theorem is considered, indeed, MacWilliams distribution provides a finite analog of the classical Gauss distribution. In particular, the MacWilliams distribution over quotient space of a code is statistical close to the uniform distribution. In the respect of lattices, the analogy of MacWilliams identity associated with nu-function was conjectured by Sole in 1995. We give an answer to this problem in positive.
Cite
@article{arxiv.2407.20874,
title = {On the MacWilliams Theorem over Codes and Lattices},
author = {Zhiyong Zheng and Fengxia Liu and Kun Tian},
journal= {arXiv preprint arXiv:2407.20874},
year = {2024}
}