Thread: Coding Theory Help Needed!

I have a few questions here that I hope you's can help me with:

1) Does a binary (8,4,5) code exist?

2)Does a binary (7,3,5) code exist?

(n,M,d) - n length of codeword, M size of code, d is the minimum distance.

3) Determine whteher or not the following subsets are supspaces of (i) Z^4 3 and (ii) [B]Z[B]^4 5 respectively.

(i) S = {x=x1x2x3x4: x1=X3 & x2=x4}

(ii) S = {x=x1x2x3x4: x1=1}

Looking forward to some replys!

1. one can easily show that such a binary code does not exist using the Griesmer bound.
2. same here.

3. I do't entirely ubnderstand what's this : (i) Z^4 3 and (ii) [b]Z[b]^4 5

I am going to rewrite what I think your 3rd question is in my notation. please let me know if I have changed it at all.

(i) Determine whether the subset S = {x = (x_1,x_2,x_3,x_4)|x_1 = x_3, x_2 = x_4} is a subspace of


(ii) Determine whether the subset S = {x = (1,x_2,x_3,x_4)} is a subspace of

For both of these you simply need to check the following properties of a subspace (since we know it is a subset):

closure under addition
closure under multiplication by a scalar
non-empty
existence of additive inverse