In this paper, we present a proof in ACL2(r) of Taylor’s formula with remainder. This important theorem allows a function f with n + 1 derivatives on the interval [a, b] to be approximated with a Taylor series of n terms centered at a. Moreover, the formula allows the error in the approximation to be bounded by a term involving the (n + 1)st derivative of f on (a, b). The results in this paper ...

Polynomial remainder codes are a large class of codes derived from the Chinese remainder theorem that includes Reed-Solomon codes as a special case. In this paper, we revisit these codes and study them more carefully than in previous work. We explicitly allow the code symbols to be polynomials of different degrees, which leads to two different notions of weight and distance. Algebraic decoding ...

In this article, we formalize that every finite cyclic group is isomorphic to a direct product of finite cyclic groups which orders are relative prime. This theorem is closely related to the Chinese Remainder theorem ([18]) and is a useful lemma to prove the basis theorem for finite abelian groups and the fundamental theorem of finite abelian groups. Moreover, we formalize some facts about the ...

For three positive integers ai, aj , ak pairwise coprime, we present an algorithm that find the least multiple of ai that is a positive linear combination of aj , ak. The average running time of this algorithm is O(1). Using this algorithm and the chinese remainder theorem leads to a direct computation of the Frobenius number f(a1, a2, a3).

The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings GR(pe, l) (including Zpe). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite chain rings. We also construct MDS self-dual...

We present a space-efficient algorithm to compute the Hilbert class polynomial HD(X) modulo a positive integer P , based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(|D|1/2+ log P ) space and has an expected running time of O(|D|1+ ). We describe practical optimizations that allow us to handle larger discriminants than othe...