Linear code with generator matrix , Mathematics

Assignment Help:

1. Consider the code of size 4 (4 codewords) and of length 10 with codewords listed below.

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 1 1 1 1

1 1 1 1 1 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1

a) Is this code a linear code? b) What is the minimum distance of the code and how many errors can the code correct? c) What is the union bound on the decoded error probability of this code when the channel is a binary symmetric channel with crossover probability p?

(The channel is also memoryless; that is any bit is in error independent of the all other bits being in error or not). d) Find a decoding rule that requires for any particular received vector y = (y1, y2, . . . , y10) two computations of the Hamming distance between two vectors of length 5 to determine which codeword was sent. (The decoding rule must be such that if the number of errors is less than the correct answer to part (b) the decoding rule will be able to correct these errors).

2. (a) The Hamming code has the following parity check matrix

747_matrix1.png

If the received vector is r = (0,1,0,1,1,1,1) find the most likely transmitted codeword (over a binary symmetric channel with error probability less than 1/2). What is the error correcting capability of the code.

(b) For the linear code with generator matrix shown below find the minimum distance of the code, the error correcting capability of the code and the code rate. Find a upper bound on the probability of a codeword decoding error on a binary symmetric channel.

2475_matrix2.png

3. Code 4 in the lecture notes (on line version) contains 32 codewords of length 15 with minimum distance 7.

(a) Simulate a communication system with this code on an additive white Gaussian noise channel. Count (at least) 100 errors and plot the error probability for signal-to-noise ratios (Eb/N0) from 0 to 6dB in steps of (no more than) 0.5dB.

(b) Determine the union bound on the performance and also plot (on the same plot as part

(a)) the union bound.

(c) Simulate the performance of a hard decision decoder that always decodes to the closest codeword. Plot the codeword error probability (on the same plot as (a) and (b)).

(d) Plot the union bound to the performance of a hard decision decoder (of part (c)).

(e) Simulate the performance of a bounded distance decoder that only corrects 0,1,2 or 3 errors. Determine the probability of choosing the wrong codeword and the probability that the received vector is not within distance 3 of any codeword (this is called a decoding failure).

(f) For a bounded distance decoder and a hard decision channel, i.e. a BSC, analyze (provide a formula) for the probability the decoder does not output the correct codeword.


Related Discussions:- Linear code with generator matrix

How long will he have to ride to burn 750 calories, Jeff burns 500 calories...

Jeff burns 500 calories per hour bicycling. How long will he have to ride to burn 750 calories? To find out the number of hours required to burn 750 calories, divide 750 throug

Prime Ideals, Given a standard 2x3 matrix show the ideal formed by the 2x2 ...

Given a standard 2x3 matrix show the ideal formed by the 2x2 minors is Prime.

Integers, hi i would like to ask you what is the answer for [-9]=[=5] grade...

hi i would like to ask you what is the answer for [-9]=[=5] grade 7

How we solve polynomial equations using factoring, How we Solve Polynomial ...

How we Solve Polynomial Equations Using Factoring ? A polynomial equation is an equation that has polynomials on both sides. Polynomial equations can often be solved by putti

How to subtract fractions with the same denominators, Q. How to Subtract fr...

Q. How to Subtract fractions with the same denominators? Ans. Subtracting fractions is basically the same as adding them. If you don't know how to add fractions, you shoul

Determine the area of the regular octagon, Determine the area of the regula...

Determine the area of the regular octagon with the following measurements. a. 224 square units b. 112 square units c. 84 square units d. 169 square units b. See

Unconditional and conditional probability, Two events A and B are ind...

Two events A and B are independent events if the occurrence of event A is in no way related to the occurrence or non-occurrence of event B. Likewise for independent

What does required to earn on his further science test in 93, Justin earned...

Justin earned scores of 85, 92, and 95 on his science tests. What does he required to earn on his further science test to have an average (arithmetic mean) of 93%? To earn an a

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd