Understanding Linear Codes for Homework

[ashnicholls]

How do you know if a code is linear.

I have only been given examples of linear codes, not what makes them linear codes.

I have to find out for homework if the product of two linear codes will be a linear code.

Eg Will C x C' be a linear code if C and C' are linear codes?

Cheers

 

[EnumaElish]

http://en.wikipedia.org/wiki/Linear_code#Formal_definition

 

[ashnicholls]

I am sure I did a search and that never came up, cheers will have a look.

 

[ashnicholls]

Yes that what I know already but don't understand,

How do I relate it to my question?

Cheers

 

[EnumaElish]

Just follow the definition. A code is linear because it is a linear subspace (with finite elements, or vectors). So you need to study linear subspaces (with finite elements).

Say C_i is a linear subspace, for each i = 1, 2 (which means that each C_i satisfies the 3 conditions of linear subspaces and has a finite number of elements). Does that mean that C = C₁ x C₂ is also a linear subspace (does C satisfy the 3 conditions of linear subspaces and have a finite number of elements, given that each of C₁ and C₂ does)?