Combinatorics: Linear Code Proof

[Askhwhelp]

Consider S ={0120, 1010, 2011} as a subset of codes of length four over Z3 with d = 3 By (a) Show that S is a linearly independent set.

I am asked to show S is a linearly independent set. However, if I add 0120 + 0120, I get 0210. Since 0210 is not in the set S, is S still a linearly independent set. If so, how could I show it?

 

[CompuChip]

Yes, the property you have in mind is called "closed" - indeed S is not closed (under the operation).
Linearly independent means that you cannot express any of the codes as a linear combination of one of the others. In other words, if you would write

$$a [0120] + b [1010] + c [2011] = [0000]$$

for integers $a$, $b$ and $c$, then the only solution to the above equation would be $a = b = c = 0$.

 

[Askhwhelp]

A follow up question:
Let S be a basis for a linear code C. How many code
words does C have? Justify your answer.

I have no clue how to go about it

 

[CompuChip]

Have you solved your original question yet, then? Because I haven't looked at it further, but I didn't give you the full answer by far.

 

[Askhwhelp]

I found a, b, c = 0...for the follow-up question, is it just 3^3?