You can not use the definition Hermitian inner product to solve this problem.

You need to use the definition of weight.

We need to generalize the equation.

1-Define the product, x*y , of two binary vectors of the same length to be the vector whose ith component is the product of the ith components of x and y.
Show that wt (x +y ) =wt (x ) +wt (y )- 2wt (x*y ).

7. The parity check matrix of a binary [n, k, d] = [15, 11, 3]
Hamming code whose columns are in numeric order is

a) Find the corresponding generator matrix.
Use the method of permuting and unpermuting the columns for the [7, 4, 3]
binary Hamming code.
b) Encode the information vector 10001000001. Explain your steps.
c) What codeword was most likely sent if the received vector is 100010101000000? Explain your steps.
d) What information vector was most likely sent if the received vector is as given in part c? Explain your steps
Note: Assume that the check bits are in columns 1, 2, 4 and 8.

Solution 62
a)
n=15
k=11
d=3
Therefore m=n-k=15-11=4

Generator G= [ Ik (kxk) P (kxm)]

Parity check Matrix H= [ PT Im]

To bring H in the standard form by permuting the rows and columns perform the following operations strictly in the order I have mentioned below.
C15 ----> C15 - C14
C14-----> C14 - C12
C13-----> C13 - C9
C12-----> C12 - C4
After this we get H in the standard form. ie. H= [ PT ...

Solution Summary

The definition of Hermitian inner products to solve this problem is determined.

Determine all binery cyclic codes of length 5.
Note: To find all cyclic codes of length n, find all ideals in B[x]/x^2+1
Note: If 1 is an Ideal (I) then R = I.
Example:
n=2
R=B[x]/x^2+1, x^2=1
R={o,1,x,1+x}
Ideals <0> = 0
<1> = R
x = (0, x, x^2...)
= (1,...

PROBLEM: For what N is it possible to list all the positive integers less than N in a Gray code, i.e., in such a way that successive numbers differ in exactly one position when the numbers are represented in binary form? For example, we can do so for N = 4 since the numbers 1, 2, 3 can be listed as 01, 11, 10 in binary form, whe

My goal, given two binary trees, is to return true if they are structurally identical, meaning they are made of nodes with the same values arranged in the same way.
Each line of values in input file "tree2Data.txt" represents one linearbinary tree, where ' _ ' represents a no value node.
Compare all trees to all other tre

Suppose that you have a dictionary whose words are not sorted in alphabetical order. As a function of the number, n of words, what is the efficiency of searching for a particular word in this dictionary ?
Do the same for dictionary whose words are sorted alphabetically. Compare results.

A. Write 3n − (k + 5) in prefix notation: ????.
b. If T is a binary tree with 100 vertices, its minimum height is ????.
c. Every full binary tree with 50 leaves has ???? vertices.

For each of the following properties, find a binary relation R such that R has that property but R^2 (R squared) does not:
(a) irreflexive
(b) antisymmetric

Demonstrate how to convert values between decimal, binary, and hexadecimal numbers.
Problem:
Using the following table, convert the defined value into the correct value for the corresponding number system:
Line Decimal Binary Hexadecimal
1 2678
2 8FC
3

4 (i) Let C be a linear code in IF. Explain what is meant when we say that C is cyclic. Give also the algebraic characterisation of cyclic codes using the ring
....
(ii) Explain why the cyclic codes in R are in 1-1 correspondence with the monic polynomials in IFq[xJ that divide ? 1. Give the definition of the generator polynom