Many linear algebra problems over the ring ZZN of integers modulo N can be solved by transforming via elementary row operations an n ⇥ m input matrix A to Howell form H. The nonzero rows of H give a canonical set of generators for the submodule of (ZZN ) m generated by the rows of A. In this paper we present an algorithm to recover H together with an invertible transformation matrix P which sat...

We discuss some problems in number theory posed by-Duro Kurepa, including the so-called left factorial hypothesis that an odd prime p does not divide 0! + 1! + · · · + (p − 1)!. 1. Introduction-D. Kurepa posed several problems in number theory that drew attention of many workers in number theory. Certainly, the most known of his problems is the so called left factorial hypothesis, which is stil...

Hopcroft’s problem in d dimensions asks: given n points and n hyperplanes in R, does any point lie on any hyperplane? Equivalently, if we are given two sets of n vectors each in R, is there a pair of vectors (one from each set) that are orthogonal? This problem has a long history and a multitude of applications. It is widely believed that for large d, the problem is subject to the curse of dime...

In this thesis we discuss how to find equivalent representations of polynomial functions over the ring of integers modulo a power of a prime. Specifically, we look for lower degree representations and representations with fewer variables for which important applications in electrical and computer engineering exist. We present several algorithms for finding these compact formulations. INDEX WORD...

This paper presents some basic theorems giving the structure of cyclic codes of length n over the ring of integers modulo pa and over the p-adic numbers, where p is a prime not dividing n. An especially interesting example is the 2-adic cyclic code of length 7 with generator polynomial X 3 + ,~X 2 + (L I)X -l, where )~ satisfies ~2 _ k + 2 = 0. This is the 2-adic generalization of both the bina...

Applying the superfast divide-and-conquer MBA algorithm for generally singular n × n Toeplitz-like or Hankel-like integer input matrices, we perform computations in the ring of integers modulo a power of a fixed prime, especially power of 2. This is practically faster than computing modulo a random prime but requires additional care to avoid degeneration, particularly at the stages of compressi...

Let $R=\mathbb{Z}_4+u\mathbb{Z}_4,$ where $\mathbb{Z}_4$ denotes the ring of integers modulo $4$ and $u^2=0$. In the present paper, we introduce a new Gray map from $R^n$ to $\mathbb{Z}_{4}^{2n}.$ We study $(1+2u)$-constacyclic codes over $R$ of odd lengths with the help of cyclic codes over $R$. It is proved that the Gray image of $(1+2u)$-constacyclic codes of length $n$ over $R$ are cyclic c...