Question

Problem 1. In this problem we work in the finite field 25, i.e. the numbers (mod 5). 1. Show that 2 is a primitive 4-th root of 1. 2. Show that X1-1= (x - 2)(x - 22)(x - 2)(X – 24). 3. Show that g(x) = (x - 2)(X - 4) generates a cyclic code C with d>3. (Hint: invoke a property that we have shown in class.) 4. What is the generating matrix G of the code C given above? 5. What are the parity-check polynomial h(x) and the parity-check ma- trix H for the code C? 6. Show that c, where c = (1,1,1,1), is a codeword in C. 7. Consider et = (0,0,2,0). What is the syndrome of e? 8. A codeword c is transmitted, but there is an error and y=(3, 2, 2, 1) is received. Find c. Problem 2. Write the elements of GF(24), obtained using the irre- ducible polynomial f(x) = 1 + X + X", using the 3 representations: poly- nomnial, binary, powers of primitive element « (where we take a = X). (In my notes on "Finite Fields", there is something similar for GF(23).) I have written in the table below the first 4 elements, you need to add the remaining 12 elements. polynomial binary o powers 0,0,0,0 0,0,0,1 X 0.0.1,0 1+X 0,0,1,1 10

Answer #1

(Sage Exploration) In class, we primarily have worked with the
field Q and its finite extensions. For each p ∈ Z primes, we can
also study the field Z/pZ , which I will also denote Fp, and its
finite extensions. Sage understands this field as GF(p).
(a) Define the polynomial ring S = F2[x].
(b) Find all degree 2 irreducible polynomials. How many are
there? For each,
completely describe the corresponding quadratic field extensions
of F2.
(c) True of false:...

G is a finite group. We have shown that C(g) ≤ G for any g ∈ G.
Regarding the cosets of C(g):
1. The elements in the same coset all have something in common that
distinguishes them
from the other cosets. Figure out what it is, state it clearly, and
prove it.
2. Find a bijection between cl(g) and the set of cosets G/C(g) =
{ aC(g) | a ∈ G }. State
it clearly and prove that it is...

Suppose f(x)=x6+3x+1f(x)=x6+3x+1. In this problem, we will show
that ff has exactly one root (or zero) in the interval
[−4,−1][−4,−1].
(a) First, we show that f has a root in the interval
(−4,−1)(−4,−1). Since f is a SELECT ONE!!!! (continuous)
(differentiable) (polynomial) function on the
interval [−4,−1] and f(−4)= ____?!!!!!!!
the graph of y=f(x)y must cross the xx-axis at some point in the
interval (−4,−1) by the SELECT ONE!!!!!! (intermediate value
theorem) (mean value theorem) (squeeze theorem) (Rolle's theorem)
.Thus, ff...

Problem 1 ...... you can use Matlan i got one so all
what i need is 2, 3 and 4 one of them or all of them ..
thanks
The following Scilab code generates a 10-second “chirp” with
discrete frequencies ranging from 0 to 0.2 with a sampling
frequency of 8 kHz.
clear; Fs = 8000;
Nbits = 16;
tMax = 10;
N = Fs*tMax+1;
f = linspace(0.0,0.2,N);
x = zeros(f);
phi = 0;
for n=0:N-1 x(n+1) = 0.8*sin(phi);
phi...

Question 1
a) To show that 3-CNF is NP-complete, we take some NP-complete
problem, say SAT, and find a polynomial-time reduction from SAT to
3-CNF. Illustrate a polynomial-time reduction that takes as input
an input F of SAT F = (¬x1 ∨ x2 ∨ x3) ∧ (x4 ⇔ ¬x5) and outputs an
input f(F) of 3-CNF so that f(F) is a “yes” instance of 3-CNF iff F
is a “yes” instance of SAT.
b) Let F be an input to...

ONLY NEED
VALUES FOR C AND D. PLUS FINAL PLOT
Use the following
code to show that the power method can be used to calculate the
largest eigenvalue and corresponding eigenvector of a covariance
matrix.
A. Generate the
data:
x <-
rnorm(100000)
dim(x) <-
c(500,200)
x <- x* 1:9
x[,1] <- x[,1]*2+
x[,3] + x[,20]
x[,5] <- x[,5]*3+
x[,3] +2* x[,20]
B.Calculate the
covariance matrix and its powers
vx <- var(x)
vx2 <- vx%*%vx
vx4 <-
vx2%*%vx2
vx8 <-
vx4%*%vx4...

ONLY NEED OUTPUT VALUES FOR C AND D. PLUS FINAL
PLOT
Use the following
code to show that the power method can be used to calculate the
largest eigenvalue and corresponding eigenvector of a covariance
matrix.
A. Generate the
data:
x <-
rnorm(1000)
dim(x) <-
c(50,20)
x <- x* 1:9
x[,1] <- x[,1]*2+
x[,3] + x[,20]
x[,5] <- x[,5]*3+
x[,3] +2* x[,20]
B.Calculate the
covariance matrix and its powers
vx <- var(x)
vx2 <- vx%*%vx
vx4 <-
vx2%*%vx2
vx8 <-...

Problem 3 you can use Matlab and also i give u the Problem 1
code its on Matlab
Using the same initial code fragment as in Problem 1, add code
that calculates and plays y (n)=h(n)?x (n) where h(n) is the
impulse response of an IIR bandpass filter with band edge
frequencies 750 Hz and 850 Hz and based on a 4th order Butterworth
prototype. Name your program p3.sce
this is the Problem 1 code and the solutin
clear; clc;...

1. Suppose we have the following relation defined on Z. We say
that a ∼ b iff 2 divides a + b. (a) Prove that the relation ∼
defines an equivalence relation on Z. (b) Describe the equivalence
classes under ∼ .
2. Suppose we have the following relation defined on Z. We say
that a ' b iff 3 divides a + b. It is simple to show that that the
relation ' is symmetric, so we will leave...

Consider a binary search tree where each tree node v has a field
v.sum which stores the sum of all the keys in the subtree rooted at
v. We wish to add an operation SumLE(K) to this binary search tree
which returns the sum of all the keys in the tree whose values are
less than or equal to K.
(a) Describe an algorithm, SumLE(K), which returns the sum of
all the keys in the tree whose values are less...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 6 minutes ago

asked 6 minutes ago

asked 27 minutes ago

asked 52 minutes ago

asked 54 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago