Gilbert-Varshamov bound
The Gilbert-Varshamov bound is like a magic rule for figuring out how many mistakes you can make when sending messages and still be able to fix them. Imagine you have to send a secret message to your friend, but sometimes the letters get jumbled up along the way. You want to make sure your friend can still read the message even if some letters are wrong. The Gilbert-Varshamov bound helps you figure out how many errors you can make and still be able to correct them.
Here's how it works: Imagine you have a certain number of letters you can use in your message. Let's say you have 10 letters to use. The Gilbert-Varshamov bound says that as long as you use those 10 letters to make a code that has enough redundancy, meaning there are extra copies of each letter, you can still fix mistakes even if up to a certain number of letters are wrong.
So if you have 10 letters, and you make a code that has enough redundancy, you can fix at least 2 mistakes. That means, if two letters get jumbled up or lost on the way to your friend, they can still figure out what you meant to say.
The Gilbert-Varshamov bound is handy because it helps you figure out how many extra copies of each letter you need to add to your code to be able to fix a certain number of mistakes. It's like a guide that tells you how to build your code so it can withstand errors and still work properly.
Here's how it works: Imagine you have a certain number of letters you can use in your message. Let's say you have 10 letters to use. The Gilbert-Varshamov bound says that as long as you use those 10 letters to make a code that has enough redundancy, meaning there are extra copies of each letter, you can still fix mistakes even if up to a certain number of letters are wrong.
So if you have 10 letters, and you make a code that has enough redundancy, you can fix at least 2 mistakes. That means, if two letters get jumbled up or lost on the way to your friend, they can still figure out what you meant to say.
The Gilbert-Varshamov bound is handy because it helps you figure out how many extra copies of each letter you need to add to your code to be able to fix a certain number of mistakes. It's like a guide that tells you how to build your code so it can withstand errors and still work properly.