Introduction To Coding And Information Theory Steven Roman | 2026 |

Mathematically, the information content ( h(x) ) of an event ( x ) with probability ( p ) is:

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. Introduction To Coding And Information Theory Steven Roman

Entropy is the average amount of information produced by a source. It is also the minimum number of bits required, on average, to encode the source without losing any information. Mathematically, the information content ( h(x) ) of

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. Entropy is the average amount of information produced

Think of entropy as the "randomness temperature." High entropy (like white noise or scrambled text) means high information density. Low entropy (like a repeating loop of silence or a predictable string of zeroes) means you can compress it down to almost nothing. Coding Theory: The Art of Reliable Imperfection If information theory is about efficiency , coding theory is about survival .

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.

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