• Home
  • Textbooks
  • Abstract Algebra with Applications
  • An Introduction to Coding Theory

Abstract Algebra with Applications

Norman J. Bloch

Chapter 15

An Introduction to Coding Theory - all with Video Answers

Educators

Chapter Questions

03:07

Problem 1

Decode the message $1,12,7,5,2,18,1,9,19,10,15,25$ given at the beginning of the chapter.

KL
Kathleen Luttrell
Numerade Educator
07:49

Problem 2

The alphabet has been shifted by an unknown number of places, and you receive the message $24,13,11,9,22,23,1,13,18$. Decode it.

Melissa Salvador
Melissa Salvador
Numerade Educator

Problem 3

Permutations may be used to provide coding schemes. For any $\alpha \in S_{26}$, we may line up the alphabet with the permuted numbers: $$ \begin{array}{ccccc} \mathrm{A} & \mathrm{B} & \mathrm{C} & \cdots & \mathrm{Z} \\ 1 \alpha & 2 \alpha & 3 \alpha & \cdots & 26 \alpha \end{array} $$
(a) Suppose that $\alpha=(1,3,5,7,9)(10,14,12)(23,21,2,8,4,16)(19,15,13)$. Decode the message $15,3,12,20,3,5,10,3,2,15$. Hint: Can you use $\alpha^{-1}$ here?
(b) Let $\beta=(1,2,3, \ldots, 25,26)$. Code the message PEACE, using $\alpha=\beta^7$ to rearrange the numbers $1,2,3, \ldots, 26$.

Check back soon!

Problem 4

ASCII code for capital letters is as indicated below: $$ \begin{array}{cccccc} \text { A } & \text { B } & \text { C } & \text { D } & \cdots & \mathrm{Z} \\ 65 & 66 & 67 & 68 & \cdots & 90 \end{array} $$
A seven bit string (seven binary digits) permits coding everything from 0 [by means of (0000000)] to 127 [by means of (1111111)], where (abcdefg) is the binary code for $a \cdot 2^6+b \cdot 2^5+\cdots+f \cdot 2^1+g \cdot 2^0$. Decode this message:
(1011010), (1000101), (1000010), (1010010), (1000001)

Check back soon!

Problem 5

(a) Find a matrix for the coding scheme $(a b c) \rightarrow(a b c a b c a b c a b c a b c)$.
(b) Discuss the error-detecting and error-correcting capabilities of the code above.

Check back soon!

Problem 6

Show that the function $\phi: \boldsymbol{Z}_2^3 \rightarrow \boldsymbol{Z}_2^4$ given by $(a b c) \phi=(a, b, a+b, a+b+c)$ is 1-1, and therefore provides a binary code.

Check back soon!

Problem 7

Show that the function $(a b c) \phi=(a+b, a+c, b+c, a+c)$ is not 1-1, therefore does not define a code.

Check back soon!

Problem 8

Discuss the error-detecting and error-correcting capabilities of the group code $\Omega: \boldsymbol{Z}_2^4 \rightarrow \boldsymbol{Z}_2^6$ whose matrix $C$ equals $$ \left[\begin{array}{llll|ll} 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 1 \end{array}\right] $$

Check back soon!

Problem 9

Same as Exercise 8 , for the code $\Gamma: \boldsymbol{Z}_2^3 \rightarrow \boldsymbol{Z}_2^7$ with matrix $C$ equal to
$$ \left[\begin{array}{lll|llll} 1 & 0 & 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 & 1 \end{array}\right] $$

Check back soon!

Problem 10

Let a polynomial code $\phi: \boldsymbol{Z}_2^3 \rightarrow \boldsymbol{Z}_2^5$ have the multiplier $G(x)=1+x^2$. That is, $(a b c) \phi=$ the coefficient string of the polynomial product $\left(a+b x+c x^2\right)\left(1+x^2\right)$. Compute each of the following code words.
(a) (101) $\phi$
(b) $(011) \phi$
(c) $(001) \phi$

Check back soon!

Problem 11

Let $\mathbf{Y}$ and $\mathbf{Z}$ be elements of $\boldsymbol{Z}_2^n$. Prove that $\mathbf{Y}=\mathbf{Z}$ iff $\mathbf{Y}+\mathbf{Z}=\mathbf{0}$.

Check back soon!

Problem 12

Let a certain group code have the matrix
$$ \left[\begin{array}{lll|ll} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 \end{array}\right] $$
Follow the method of Example $15-5$ to decode the following received messages.
(a) (10101)
(b) $(11100)$
(c) $(00101)$

Check back soon!

Problem 13

Let a group code $\phi$ have the matrix
$$ \left[\begin{array}{lll|lll} 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 & 0 \end{array}\right] $$
(a) List the eight possible syndromes $\mathbf{S}_i$ for $\phi$.
(b) For as many $\mathbf{S}$ as possible, find a string of weight 1 or less whose syndrome equals $\mathbf{S}_i$, to serve as coset leader.
(c) For the remaining $\mathbf{S}_i$, find coset leaders of weight 2 .
(d) Follow the method of Example 15-6 to decode each of the following received words:
$$ (101 \underline{010}),(111110),(010011),(011000) $$

Check back soon!

Problem 14

A certain group code has the matrix
$$ \left[\begin{array}{l|lll} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 0 \end{array}\right] $$
Follow the method of Example 15-6 to decode each of the following:
(11111101), (00110001), (10011101), (00010010)

Check back soon!

Problem 15

A certain code $\phi: \boldsymbol{Z}_2^2 \rightarrow \boldsymbol{Z}_2^6$ has the image $\{(000000),(101110),(110101),(011011)\}$.
(a) Show that $\phi$ is a group code.
(b) Discuss the error-detecting and error-correcting capabilities of $\phi$.

Check back soon!

Problem 16

Answer the following questions for the polynomial code of Exercise 10.
(a) Decode the message (11011). Hint: See the remark about long division at the end of the chapter.
(b) The message (01111) was received. Was the message transmitted correctly?
(c) For each of the received messages that follows, test to detect errors and decode if no error is detected.
(i) $(01010)$
(ii) $(10110)$

Check back soon!

Problem 17

Define a polynomial code $\Omega: \boldsymbol{Z}_2^4 \rightarrow \boldsymbol{Z}_2^7$ by means of a "multiplier" $G(x)=1+x+x^3$. For each of the following received messages, test for errors and decode if no error is detected.
(a) $(0011010)$
(b) (1101101)
(c) $(0000000)$
(d) $(0101111)$

Check back soon!

Problem 18

Same as Exercise 17, except that $G(x)=1+x^2+x^3$ and the received messages are
(a) $(1011000)$
(b) (1001110)
(c) $(0101001)$
(d) (1111111)

Check back soon!

Problem 19

Suppose that a binary group code has matrix $\left[I_{5 \times 5}: P_{5 \times 4}\right]$. That is, the identity matrix part $I$ is $5 \times 5$, and the parity check matrix $P$ is $5 \times 4$ (five rows, four columns).
(a) How many different code words are there for this code?
(b) How many different syndromes would there be?
(c) How many different zero- and one-error patterns could be corrected? On top of those, how many two-error patterns could be corrected?
(d) How many different error patterns would go uncorrected, or wrongly corrected?

Check back soon!
10:39

Problem 20

Suppose that a transmission channel is such that the probability of an error on any one transmitted digit is $p=0.001$, and that errors from digit to digit are independent of each other.
(a) Show that the probability of two or more errors in a string of seven transmitted digits is approximately 0.00002093 . Hint: Use the binomial probability distribution, with $p=0.001$.
(b) Show that the probability of some errors in four transmitted digits is approximately 0.003994 .

Chris Trentman
Chris Trentman
Numerade Educator
10:39

Problem 21

Suppose that the probability of an error in each transmitted digit is $p=10^{-5}$.
(a) Find the probability of two or more errors in a string of seven transmitted digits.
(b) Find the probability that a string of four digits is transmitted with zero errors.
(c) By what factor does a $(7,4)$ binary code that corrects all one-error messages reduce the proportion of incorrectly decoded messages?

Chris Trentman
Chris Trentman
Numerade Educator

Problem 22

Show that if $\phi: \boldsymbol{Z}_2^k \longrightarrow \boldsymbol{Z}_2^n$ is a homomorphism, then there exists a $k \times n$ matrix $C$ such that $\mathbf{X} \phi=\mathbf{X} C$ for all $\mathbf{X} \in \boldsymbol{Z}_2^k$. Hint: Consider the matrix whose rows are $(1000 \cdots 00) \phi,(01000 \cdots 00) \phi,(00100 \cdots 00) \phi, \cdots,(00 \cdots 0001) \phi$.

Check back soon!

Problem 23

Can there exist a group code $\phi: \boldsymbol{Z}_2^k \rightarrow \boldsymbol{Z}_2^\pi$ such that $\phi$ is not a homomorphism? Explain.

Check back soon!