Linar Codes and cacluating the cosets

[Rubik]

Given a liear code generated by 01111, 11010 and 10100 how do you calculate the cosets of C? Does this mean because it is generated by that matrix that it is not my acutal code C so am I suppose to find C then find my cosets or do I straight up use the generated matrix as it is equivalent to C?

 

[micromass]

Let's make some assumptions here: you're working in the field $$\mathbb{F}_2$$, and thus your codes are in $$\mathbb{F}_2^5$$.

So, C is just a subspace of $$\mathbb{F}_2^5$$, and you need to find all x+C, for $$x\in \mathbb{F}_2^5$$.

The way I would do this is to complete the basis of C to a basis of $$\mathbb{F}_2^5$$. Thus 01111, 11010 and 10100 are already 3 linear independent vectors. You'll need to find 2 other vectors x and y such that $$\{01111,11010,10100,x,y\}$$ form a basis for $$\mathbb{F}_2^5$$.

It can then easily be shown that

$$\{(\alpha x+\beta y)+C~\vert~\alpha,\beta\in \mathbb{F}^2\}$$

are the cosets of C. Thus there are 4 cosets!

 

[Rubik]

Sorry yes those assumptions are correct but also I made a slight mistake in my third term my code C = {01111, 11010, 10101} does this now mean my vectors are no longer lineraly independent?

 

[micromass]

Hmm, I indeed fear those vectors are not linear independent, since 01111=11010+10101.
So a basis would consist of {11010,10101}. Now you can do the same thing. Add 3 vectors to complete this to a basis of $$\mathbb{F}_2^5$$...

 

[Rubik]

Okay thanks I will give it a go.. I am even more appreciative of this advice seeing as it is coming from a Pink Floyd fan! :D