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# Powerpoint (418) | Mathematics homework help

Multimedia PresentationIn this assignment, you will create a correctable code for a list of key words. Your task is to create an efficient, correctable code for a list that contains at least 6 key words. The words in your code will be represented as binary strings using only 0 s and 1 s. Stringent correctability requirements mean your code must have a minimum distance of 3.
First, watch the following two videos, then read the following information to understand the definitions for bit, binary word, code, codewords, Hamming distance, and minimum distance of a code:
Video 1: Parity Checksums
Video 2: Hamming distance
Consider a sequence of 0 s and 1 s of length n. This can be represented by an n-tuple of 0 s and 1 s such as (1,0,1,1) if n=4. If V={0,1}, then we can form the product of V with itself n times and denote it by Vn. So Vn={(a1, a2, , an)|ai {0,1}}. Vn consists of all possible binary words of length n. We can define a metric on Vn called the Hamming distance dH as follows:
For binary words x and y of length n, dH(x, y) is the number of places in which x and y differ.
Given this metric, Vn is now a metric space, and the topology induced by this metric is the discrete topology on Vn since the topology induced by a metric on a finite set is the discrete topology, and Vn is finite.
To send a message using binary words, not all of Vn will be used; rather, only a subset of Vn will be used. A subset C of Vn is called a code of length n, and the binary words in C are called codewords. The smallest Hamming distance between any two codewords in C is called the minimum distance of the code C.
It turns out that, if a code C of length n is designed so that the minimum distance of C is d, then any binary word that had up to d-1 errors can be detected. Furthermore, any binary word that had floor((d 1)/2) or fewer errors can be corrected. [Here, floor is the floor function; for example, floor(3.6)=3 and floor(8)=8.]
Now, you re ready to create your correctable code. Create a code consisting of binary codewords. The code must meet three requirements Contain at least 6 codewords Have a minimum distance of 3 (explain why a min distance of 4 is no better than 3) Maintain efficiency by using the fewest number of bits per codeword as possible Clearly document and describe your code: what it is, why you chose it, etc. Discuss how topology relates to the selection of your code and the Hamming metric
A few notes about format: use MS PowerPoint for your presentation; develop a presentation that is 10-15 slides in length; incorporate audio files into your presentation in order to explain your work; use Equation Editor for all mathematical symbols, e.g. x X or Cl(A) Cl(X-A); and select fonts, backgrounds, etc. to make your presentation look professional.
Course and Learning ObjectivesThis Writing Assignment supports the following Course and Learning objectives:CO-4 Determine if a topological space is a metric space and generate a topology from a metric.LO-13: Understand the definitions of a metric and metric space.LO-14: Develop a topology from a metric.

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