Let Ax = B be a system of m x n linear equations with integer coefficients. Assume the rows of A are linearly independent and denote by X (respectively Y) the maximum of the absolute values of the m x m minors of the matrix A (the augmented matrix (A, B)). If the system has a solution in nonnegative integers, it is proved that the system has a solution X =… (More)
1. Introduction. Sometimes one is interested in exact solutions of linear equations and cannot tolerate any errors at all, be it round-off errors, truncation errors or otherwise. In such situations division cannot be used. Eliminating variables by cross-multiplying leads to a tremendous growth in the size of intermediate results. This is well illustrated in… (More)
We study categories of codes and precodes. The objects in these categories capture the encoding and decoding process of error control codes, source codes, or cryptographic codes. We show that these categories are complete and cocomplete. This gives a wealth of new code constructions.
Blakley and Borosh introduced a general theory of codes, encompassing cryptographic and error control codes among others. They explored the properties of such general codes with methods from rela-tional algebra and set theory. We provide a categorical point of view, which leads to new constructions of codes. We also exhibit a Jordan-Hölder type theorem and… (More)
WSPC/Book Trim Size for 9in x 6in justchapter7 Preface The main justification for this book is that there have been significant advances in continued fractions over the past decade, but these remain for the most part scattered across the literature, and under the heading of topics from algebraic number theory to theoretical plasma physics. We now have a… (More)