# Powerpoint (418) | Mathematics homework help

**Multimedia Presentation**

In 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 Objectives**

This 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.