Please solve this question:
1. Find the Hamming weight of each codeword in Table 31.1.
2. Find the Hamming distance between the following pairs of vectors: {1101, 0111}, {0220, 1122}, {11101, 00111}.
3. Referring to Example 1, use the nearest-neighbor method to decode the received words 000010 and 1110100.
4. For any vector space V and any U, W in Fn, prove that the Hamming distance has the following properties:
a. d(u,v) = d(v, u) (symmetry)
b. d(u, v) = 0 if and only if u = v
c. d(u,v) = d(u + W,v + w) (translation invariance).
5. Determine the (6, 3) binary linear code with generator matrix:
1 0 0 0 1
0 1 0 1 0
0 0 1 1 0
6. Show that for binary vectors, wt(u + v) ≤ wt(u) + wt(v) and equality occurs if and only if for all i the ith component of u is whenever the ith component of v is 1.
7. If the minimum weight of any nonzero codeword is 2, what can we say about the error-detecting capability of the code?
8. Suppose that C is a linear code with Hamming weight 3 and that C' is one with Hamming weight 4. What can C do that C' can't?
9. Let C be a binary linear code. Show that the codewords of even weight form a subcode of C. (A subcode of a code is a subset of the code that is itself a code.)
10. Let C = {0000000, 1110100, 0111010, 0011101, 1001110, 0100111, 1010011, 1101001}. What is the error-correcting capability of C? What is the error-detecting capability of C?