Algorithms and Implementation for P-adic Cyclic Codes Using Exact Arithmetic Library Developed for Quantum Computing

Abstract

Abstract : The first part of the research is that we have expanded the Exact Scientific Computational Library (ESCL), and Dixon's algorithm on rational N by N matrix inverse was implemented. We studied and experimented the relation of required length M of p-adic expansion and the prime p, and the possible use of the length of periodicity of a rational number's p-adic expansion in determining the length of required M in rational matrix operations. The second part of the work is to develop and implement computational algorithms for p-adic cyclic code generation, which is based on the results of the paper, Modular and p-adic cyclic codes, by A.R. Calderbank and N.J.A. Sloane. Algorithms and software have been developed to give an alternative solution to factorize the polynomial X-1 over the ring of integers modulo p(a), where p is a prime not dividing n, and it can generate the table of cyclic codes using the divisors of X-1 as their generator polynomials. All the implementation of ESCL is in C++, as well as the software to generate p-adic cyclic codes.