Reference no: EM131213417
We can use the 2n different binary strings of length n to code (i.e., uniquely label) 2n distinct objects. However, some pairs of these 2n objects will have codes that differ only in one position. Thus, if we mistype even a single bit of an object's code we would inadvertently specify a different object than the one intended. For example, suppose n = 4 and the code for "Apple iPhone" is 0110 while the code for "Samsung Galaxy" is 0010. If we wanted to order the iPhone through a web form but mistyped the second bit of its code, we would receive the Galaxy instead! To avoid this, we would like the objects to be coded in such a way that no two of them have codes that differ in only one position. In that case, if we make only one typing error in entering the code, the system could inform us that the code we entered is invalid, instead of mistaking it for the code of a different object. Coding schemes that have this property are called "error-detecting codes".
The question now arises: Using binary strings of length n, how many different objects can we label in such a way that there are no two objects whose codes differ in only one position? In this question, you will show that the answer is 2n-1.
a. Let n be any positive integer, and let S be any set of binary strings of length n such that no two strings in S differ in only one position. Prove that S contains no more than 2n-1 strings.
b. Prove that for every positive integer n, there exists a set S of binary strings of length n that contains 2n-1strings no two of which differ in only one position. (There are multiple ways to prove this, not all of which involve induction. You need only give one proof, which may or may not use induction.)
|
Why were significant fears about the impact of freed slaves
: Introduce the debate surrounding abolition vs. equality and how it led to the American Civil War. Why were there significant fears about the impact of freed slaves in the North? Who had the most to win/lose?
|
|
Management characteristics or job characteristic
: Dissertation on "To determine whether it is management characteristics, or job characteristic, that have the greatest influence on engagement in a Liberian insurance company"
|
|
Networking models used for communications
: Assuming you resolved the beginning of this chapter's modeling activity, how would you compare the following steps taken to design a communications system to the networking models used for communications?
|
|
Design a voltage divider to provide the approximate
: The number of resistors, their values, and their wattage ratings must be specified. A schematic showing the circuit arrangement and resistor placement must be provided.
|
|
Determining the binary strings of length
: Prove that for every positive integer n, there exists a set S of binary strings of length n that contains 2n-1strings no two of which differ in only one position. (There are multiple ways to prove this, not all of which involve induction. You need ..
|
|
Implement a selection sort
: Implement three functions whose prototypes are given - You are going to use a static array declared using { 12, 1, 6, 8, 5, 9, 22, 9, 13, 17 } to initialize an STL list, as shown in the slides.
|
|
Identify and describe a current issue in american government
: Identify and describe a current issue (this could be a controversy or just an issue of importance) in American Government. Explain in 1-2 pages how one of the modern philosophers we've studied would respond to that issue.
|
|
What do we know about the mitzvah of giving charity
: What is the source for the Mitzvah of Giving Charity in Jewish Tradition? Based on this week's lecture, what do we know about the Mitzvah of Giving Charity? How do we know?
|
|
What is the capital balance of the remaining three partners
: Assume that the partners share profits and losses 4:3:2:1, respectively. Pnda retires and is paid $305,000 based on the terms of the original partnership agreement. If the bonus method is used, what is the capital balance of the remaining three partn..
|