China
Africa
Explore chapters and articles related to this topic
The last chapter
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
Each element of GR(m, Z4) = Z4[X]/(h(X)) is uniquely represented by a polynomial of degree < m. It follows that the Galois ring R = GR(m, Z4) has 4m = q2 elements. If h(X) is the Hensel lifting of a primitive polynomial, a zero ∊ of h(X) is an element of order q − 1. Denote by the set of all powers of ∊ together with 0. Observe that is multiplicatively closed. Then |T | = q. The zero divisors of R are precisely the multiples of 2. They form the ideal 2R, which is a maximal ideal, and we have that the factor ring, which of course is a field, is the finite field with q elements: R/2R ∼ = Fq. The q2 − q = q(q − 1) elements not in 2R are units. Each element of the form 1 + 2r has order 2. It follows that 1 + 2R is an elementary abelian group of order q. This clarifies the structure of the group of units: it is the direct product of the cyclic group of order q − 1 generated by ∊ and the elementary abelian group 1 + 2R.
Ideal Decomposition in Number Fields
Published in Richard A. Mollin, Algebraic Number Theory Second, 2011
for some m∈N. Also, V0/V1 is a cyclic group, and Vj-1/Vj is an elementary abelian p-group by parts (e)-(f) just proved, so V1 is a p-group. Hence, TP(1)(K/F) is the maximal tamely ramified extension at P contained in K, which is (g).
Using Individual Factor Information in Fractional Factorial Designs
Published in Technometrics, 2019
William Li, Robert W. Mee, Qi Zhou
Two-level fractional factorial designs are widely used to investigate factorial effects. Traditional work on experimental design has focused on constructing effective and efficient designs, while little attention has been paid regarding how to assign columns to factors after a design is selected. This motivated Li, Zhou, and Zhang (2015) to propose the individual word length pattern (iWLP) to measure aliasing severity of columns for regular two-level factorial designs. A regular design is one whose columns form an elementary Abelian group, and any two effect contrasts are either orthogonal or fully aliased. For a nonregular design, many factorial effects will be partially aliased. Partial aliasing has two consequences: bias to estimates in the fitted model when active interactions are omitted and a decrease in precision due to correlated contrasts for aliased terms included in the fitted model. In the first part of this article we consider nonregular designs and propose a new measure, the individual generalized word length pattern (iGWLP), using an idea similar to Tang and Deng’s (1999) G2-aberration. While nonregular designs have more complicated aliasing structures, they also provide more run size options than do regular designs. The proposed iGWLP criterion can be used to reveal additional information on aliasing between effects involving an individual factor and other effects, as illustrated by the following example.