Linear Code

A Binary Code is linear if

2. if then

Rank of a Linear Code
A code of length and rank is called an code.
If a linear code has minimum distance then it is a code

Say has a basis
Then so
i.e. an code is an code

The Information Rate is

We define

Ring over binary words

Hamming weight
Parity Check Code
Dual Code
Generator Matrix
Parity Check Matrix
Syndrome
Syndrome Decoding

Hamming Codes
Reed-Muller Codes

Cyclic Code
Defining Set (wasn’t done in lectures)
Cyclic Code with Defining Set
BCH Code