博客

School of Mathematics and Statistics

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 Zⁿ ₂with 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 ⁿ ₂we 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 ⊆ Z⁶₂ be 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 · 2^k−1 .

更多代写：Assignemnt代写  托福家庭版  Accountant会计网课代上价格  英国essay范文  澳大利亚研究性论文代写  MATH数学作业代写

合作平台：essay代写 论文代写 写手招聘 英国留学生代写