Introduction To Coding And Information Theory Steven Roman Apr 2026

When your data corrupts, you are witnessing a violation of the Hamming distance. When your compression algorithm bloats instead of shrinks, you are witnessing low entropy.

[ h(x) = -\log_2(p) ]

Mathematically, the information content ( h(x) ) of an event ( x ) with probability ( p ) is: Introduction To Coding And Information Theory Steven Roman

When most people hear the word "code," they think of spies, secret languages, or JavaScript. When they hear "information," they think of news or data. But in the mathematical universe, these two concepts are married in a beautiful, rigorous dance that underpins every text message, every streaming video, and every photograph from Mars.

If I tell you something you already know (e.g., "The sun will rise tomorrow"), I have transmitted very little information. If I tell you something shocking (e.g., "The sun did not rise today"), I have transmitted a massive amount of information. When your data corrupts, you are witnessing a

In Shannon’s world,

If you receive a 7-bit string, you run the parity checks. The result (called the syndrome) is a binary number from 001 to 111. That number tells you exactly which bit to flip to fix the message. When they hear "information," they think of news or data

[ H = -\sum_{i=1}^{n} p_i \log_2(p_i) ]