Algebraic coding theory- Golay Code

[hatsu27]

Homework Statement

does anyone know why C24 (the extended Golay Code) doesn't have any words of weight 20? I know that it only has words of weight 0,8,12,16, & 24, but why is 20 skipped here?

Homework Equations

I am asked to deduce this from the fact after I have shown that the code does contain the word of all one's. I did this by showing how it is a linear combination of all 12 rows in the generator matrix [I,B] since each column has odd weight. But I am not sure how this is connected to the weights of all the words in C24.

The Attempt at a Solution

Now I was thinking that since the only weight of words in G are either 8 or 12 then any linear combination of the words would be multiples of 8 or 12, but I don't really see why that would be and just me wishful thinking since I have been mulling this question over for 2 days and everywhere I think it out I run into walls. Any ideas?

 

[haruspex]

Not an area I know anything about, but I note that 4 is also skipped. Do you know why that is? Is there perhaps some symmetry between case N and case 24-N?

 

[hatsu27]

the smallest distance for the code is 8 so that is the smallest weight possible. That means that the words must be separated by a distance of 8. The Golay is also a double self-dual code so all weights must be divisible by 4

 

[haruspex]

the smallest distance for the code is 8 so that is the smallest weight possible. That means that the words must be separated by a distance of 8. The Golay is also a double self-dual code so all weights must be divisible by 4

Ok, so how about a possible symmetry? Would the existence of a weight N imply the existence of a weight 24-N?