You add more information to make sure you don't make mistakes. This is where you send the y- value that matches the x-coordinate. There is an error if the points don't fall on the same line. To figure out where the error is, you just send one more value, instead of six.
The bigger the message, the better. Let's say you want to send a long message. You only need 1,001 numbers to find the error and 1,002 to correct it if you use the code that interpolates a polynomial through points. Adding more points can help identify and correct errors. The efficiency between the two codes decreases as the length of your message increases.
A Reed-Solomon code is more efficient. Calculating more errors with greater efficiency has been achieved since 1960. A mathematician and computer scientist at the University of Toronto said it was clean and elegant. In half an hour, it can be taught to a second-year undergrad.
Reed-Solomon codes can be used to store and transmit information. The same concept has been used in other areas.
If you want to distribute a secret among several parties, no one person can access the whole secret, but together they can. Imagine a missile launch code. In order to distribute the results to different people, you have to decode the numbers in a polynomial, evaluate it at a set of points, and give each result to a different person.
Reed-Solomon codes have been used in a number of areas. You need to verify that the computation you get back is correct when you run a computation that is too complicated for your laptop. If the cluster hasn't done the computation right, you can ask for more information. Jade Nardi is a research fellow at the mathematics institute of Rennes in France. The way it relies on these codes blows my mind.
The Reed-Solomon codes have a constraint. You can only evaluate your polynomial at a fixed set of values, because they are constructed in such a way. You can only use a set of numbers to decode your message. The bigger the alphabet, the longer the messages you can send, and the more power you need to decode them.
The mathematicians wanted an even better code.
A more powerful code would allow you to store longer messages without increasing the size of your alphabet. In order to do this, mathematicians devised codes that involve interpolating a function through points on a curve. The code that came out of nowhere is better than any other code we know how to make. This is the best thing that has happened. It was a big surprise.
There is a single problem. It is easier to implement a Reed-Solomon code than it is to implement an algebraic geometry code. This is state-of-the-art, but it is still being investigated to see if it will turn into something practical. It is difficult to handle these codes on a computer.
In real world applications, Reed-Solomon codes and related forms of error correction are enough. That may not be the case sometimes. If quantum computers are available in the future, they will be able to break today's cryptography protocols. Researchers have been looking for schemes that are resistant to quantum attacks. Reed-Solomon codes are not strong enough to be a top contender. It's possible that certain versions of geometry codes will work. Some researchers think that the role of algebraic geometry codes in cloud computing could be positive.
Elena Berardini is a researcher at the University of Technology in the Netherlands who works on algebraic geometry codes. It might be useful for something completely unforeseen after 50 years.
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