Why does Code 3 fail the prefix-free condition?

Code 3 fails the prefix-free condition because specific codewords act as prefixes for other codewords within the same code. Specifically, the codeword 0 assigned to

x_1

is a prefix of the codeword 00 assigned to

x_3

, and the codeword 1 assigned to

x_2

is a prefix of the codeword 11 assigned to

x_4

Conditions

Use the Code 3 column of the displayed table.
Apply the prefix-free definition: no codeword is a prefix of another.

Reasoning, step by step

Extract the codewords from Code 3: $x_1=0$ , $x_2=1$ , $x_3=00$ , $x_4=11$ .
Check if 0 is a prefix of 00. Yes, 00 begins with 0.
Check if 1 is a prefix of 11. Yes, 11 begins with 1.
Conclude that Code 3 violates the prefix-free condition due to these prefix relations.

Example

The slide explicitly marks Code 3 as 'not prefix-free' and highlights that 0 is a prefix of 00 and 1 is a prefix of 11, causing potential decoding confusion.

Common misconceptions

Assuming that because the codewords are distinct (0, 1, 00, 11 are all different), the code must be prefix-free.
Thinking that prefix-freeness only applies to fixed-length codes.

Watch the explanation

YouTubeClassification of Codes | Source coding | Information Theory and Coding

4:35 – 5:04Watch this moment ↗

Explore next

Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.