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Two Decades of Multidimensional Systems Research and Future Trends
Published in Krzysztof Gałkowski, Jeff David Wood, Multidimensional Signals, Circuits and Systems, 2001
When the principal ideal ring is a Euclidean domain. Euclid's algorithm can be used to construct £>„. as described in the unimodular matrix completion strategy in (Newman 1972). From Fact 1.2. it follows that the elements in the top row of a unimodular matrix are relatively prime. For convenience in later usage, the procedure for constructing a unimodular matrix from a specified top row will be called matrix completion. Furthermore, the following result is know n to hold.
Cyclic codes
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
As everybody knows the Euclidean algorithm is a fast method to compute the greatest common divisor (a, b) of two elements a, b of a Euclidean domain. The Euclidean domain R comes with a norm N and the basis of the Euclidean algorithm is division with remainder a = q · b + r,
Integral Domains, Ideals, and Unique Factorization
Published in Richard A. Mollin, Algebraic Number Theory Second, 2011
When D possesses a Euclidean function then D is called a Euclidean domain.
Fully homomorphic encryption: a general framework and implementations
Published in International Journal of Parallel, Emergent and Distributed Systems, 2020
An example of an Euclidean domain is the set of integers, with valuation function for If one assumes moreover that the remainder is nonnegative, then division has unique quotients and remainders. An example of an Euclidean domain where division has unique quotients and remainders (without additional hypothesis) is the set of polynomials in one undeterminate t with real coefficients, with valuation function the degree of p. On the other hand, is not an Euclidean domain: one can only divide by polynomials whose leading coefficient is 1 or −1, but then, the quotient and the remainder are unique.