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1. Give an example of one codeword c E F. Explain how you know it is a codeword. (Note, provide a different codeword than any codewords given in the next question). 2. Determine whether each of the following vectors is a codeword. If it is a codeword, find a solution x such that Gx = c. If not, explain how you know. (Use and augmented matrix. Show all work) (a) C₁ = 0 0 1 6 0 (b) C₂ = 0

1. Give an example of one codeword c E F. Explain how you know it is a codeword. (Note, provide a different codeword than any codewords given in the next question). 2. Determine whether each of the following vectors is a codeword. If it is a codeword, find a solution x such that Gx = c. If not, explain how you know. (Use and augmented matrix. Show all work) (a) C₁ = 0 0 1 6 0 (b) C₂ = 0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I have no idea where to even start with this practice. I do not know how to get a codeword or what it looks like. Please help with part a and b.

• We can write that C = {c such that c is a codeword}. • A specific vector c in F is considered a codeword if it is in the code C (i.e., an element of the set C). Otherwise, it is not a codeword. %3D • We will consider a basis of the subspace C.We will take the basis vectors of C and combine them to form a matrix G that we call the generator matrix of the code. • Example: 0 0 0 1 0 1 0 0 0 0 0 is a generator matrix for the code The matrix G = 0 1 1 1 10 1 1 1 10 1 C = {c where c= x1 + x2 + 13 + x4 19-0-0-0 Exercises (part A) 1. Give an example of one codeword c E F,. Explain how you know it is a codeword. (Note, provide a different codeword than any codewords given in the next question). 2. Determine whether each of the following vectors is a codeword. If it is a codeword, find a solution x such that Gx = c. If not, explain how you know. (Use and augmented matrix. Show all work) 0. 1. (a) c1 = %3D 0. %3D 0. 1. 1. 0. 2.
Transcribed Image Text:• We can write that C = {c such that c is a codeword}. • A specific vector c in F is considered a codeword if it is in the code C (i.e., an element of the set C). Otherwise, it is not a codeword. %3D • We will consider a basis of the subspace C.We will take the basis vectors of C and combine them to form a matrix G that we call the generator matrix of the code. • Example: 0 0 0 1 0 1 0 0 0 0 0 is a generator matrix for the code The matrix G = 0 1 1 1 10 1 1 1 10 1 C = {c where c= x1 + x2 + 13 + x4 19-0-0-0 Exercises (part A) 1. Give an example of one codeword c E F,. Explain how you know it is a codeword. (Note, provide a different codeword than any codewords given in the next question). 2. Determine whether each of the following vectors is a codeword. If it is a codeword, find a solution x such that Gx = c. If not, explain how you know. (Use and augmented matrix. Show all work) 0. 1. (a) c1 = %3D 0. %3D 0. 1. 1. 0. 2.
3. • We call a matrix H the parity of the code when the is the matrix of all It also for the parity check H that for any c E C der the matr A=-2 Parity check matrix ind a ba • Example: 0 0 0 1 0 0 00 - Given the generator matrix G = we can find the parity check matrix %3D 1 1 0 1 10 1 1 1 1 0 1 and 4th 1 0(N0 10 1 100 1 1 1 1 1 0 ) 9 9 Fi 194 Exercises (part B) 0. Ose the parity check matrix H to determine whether each of the following is a code word. Show all multiplications, and state and explain your conclusion. 4. Verify that H is a parity check matrix for G. Show all multiplications. 0. (b) c2 = (a) c1 = %3D 1. 1. 0. 6. Read slides 1/37-7/37, and 15/37 of https://www.fi.muni.cz/usr/gruska/crypto11/crypto_ 01 2x2.pdf. Write a brief reflection. (What did you find interesting in this reading? What did it make you wonder?) no.
Transcribed Image Text:3. • We call a matrix H the parity of the code when the is the matrix of all It also for the parity check H that for any c E C der the matr A=-2 Parity check matrix ind a ba • Example: 0 0 0 1 0 0 00 - Given the generator matrix G = we can find the parity check matrix %3D 1 1 0 1 10 1 1 1 1 0 1 and 4th 1 0(N0 10 1 100 1 1 1 1 1 0 ) 9 9 Fi 194 Exercises (part B) 0. Ose the parity check matrix H to determine whether each of the following is a code word. Show all multiplications, and state and explain your conclusion. 4. Verify that H is a parity check matrix for G. Show all multiplications. 0. (b) c2 = (a) c1 = %3D 1. 1. 0. 6. Read slides 1/37-7/37, and 15/37 of https://www.fi.muni.cz/usr/gruska/crypto11/crypto_ 01 2x2.pdf. Write a brief reflection. (What did you find interesting in this reading? What did it make you wonder?) no.
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