Mathematics代写

## MT4516 Finite Mathematics

Mathematics代写 Consider a code C ⊆ Z n 2 with minimum distance at least 2k + 1. For w ∈ C, let B(w, k) be the Hamming ball with centre w and radius k.

### 1.The set Zn2with the Hamming distance d is a metric space. In other words, d has the following properties:  Mathematics代写

(i) d(x, y) 0;

(ii) d(x, y) = 0 ⇐⇒ x = y;

(iii) d(x, y) = d(y, x);

(iv) d(x, z) d(x, y) + d(y, z) (the triangle inequality);    Mathematics代写

for all x, y, z Zn 2

2.Consider a code C Z n 2 with minimum distance at least 2k + 1. For w C, let B(w, k) be the Hamming ball with centre w and radius k.

Prove that, for distinct w1, w2 C, we have

### 3.Prove that for any x, y ∈ Z n2we have

wt(x) = d(x, 0)    and  wt(x + y) wt(x) + wt(y).

4.Prove that if x, y Z n 2 both have even weights then their sum x + y also has even weight.

### 5.For each of the following codes, fifind the minimum distance of the code and determine how many errors it can detect and correct.

(i) C1 = {0000, 1100, 1010, 1001, 0110, 0101, 0011, 1111} in Z42;

(ii) C2 = {10000, 01010, 00001} in Z 52 ;

(iii) C3 = {000000, 101010, 010101} in Z62.

Which of these codes can be extended by a further codeword without changing the minimum distance?

6.Construct a code in Z62 which can encode fifive messages and correct one error.

### 7.Consider the code  Mathematics代写

C = {000000000, 100110110, 010011011, 001101101,

110101001, 101011101, 011110010, 111000111}.

Prove that C is not linear.

Determine the weights of the code-words.

Verify that the minimum distance is 3.

What are the error-detecting and error-correcting capabilities of C ?

8.Let C Z72be the linear code with the generator matrix

Find a parity check matrix for C , list all the code-words, and determine the error-detecting and error-correcting capabilities of C .

### 9.Let C ⊆ Z62be the linear code with the parity check matrix  Mathematics代写

Find a generator matrix for C

list all the code-words, and determine the error-detecting and error-correcting capabilities of C.

10. Let C Zn2 be a linear code with a generator matrix G of dimension k × n . For each t ∈ {1, . . . , n} defifine Ct = {x1x2 . . . xn C : xt = 0} .

(i) Prove that each Ct is a linear code.  Mathematics代写

(ii) Prove that either |Ct| = |C| or else |Ct| = |C|/2 .

(iii) Prove that if G has no column consisting entirely of zeros, then the sum of the weights of all of the code-words in C is equal to n · 2k1 .