Upper and lower bounds for maximum number of codewords

Error

There was an error loading the page.

Contributors

Christian Lawson-Perfect

Feedback

From users who are members of Christian's workspace :

Christian Lawson-Perfect

said

Ready to use

10 years, 1 month ago

History

○

Christian Lawson-Perfect 10 years, 1 month ago

Gave some feedback: Ready to use

○

Christian Lawson-Perfect 10 years, 6 months ago

Created this.

There is only one version of this question that you have access to.

There is one other version that you do not have access to.

Question statement

Using the Hamming, Singleton and Gilbert–Varshamov bounds, give upper and lower bounds for the maximum number of codewords in the following codes:

Give any introductory information the student needs.

			Name	Type	Generated Value

field_sizes	list	[ 4, 5, 6 ]
code_lengths	list	[ 10, 15, 5 ]
min_distances	list	[ 2, 2, 3 ]
hamming_bounds	list	[ 1048576, 30517578125, 1296 ]
singleton_bounds	list	[ 262144, 6103515625, 216 ]
upper_bounds	list	[ 262144, 6103515625, 216 ]
lower_bounds	list	[ 95326, 1907348633, 486 ]
errors_corrected	list	[ 0, 0, 1 ]

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
shuffle([2,3,4,5,6,7])[0..3]

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `list`

[ 4, 5, 6 ]

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
Part b) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
Part c) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

A code over $\mathbb{Z}_{\var{field_sizes[0]}}$ with $n = \var{code_lengths[0]}$ and $d = \var{min_distances[0]}$ .

Gap 0. $\leq M \leq$ Gap 1.

{lower_bounds}

{hamming_bounds}

{errors_corrected}

{singleton_bounds}

Advice

The Gilbert-Varshamov bound gives a lower bound for $M$ . The Hamming and Singleton bounds give upper bounds, so use the lowest of those.

a)

Gilbert-Varshamov: $M \geq \frac{\var{field_sizes[0]^code_lengths[0]}}{\var{sum(map(comb(code_lengths[0],j),j,0..(min_distances[0]-1)))}} \approx \var{precround((field_sizes[0]^code_lengths[0])/sum(map(comb(code_lengths[0],j),j,0..(min_distances[0]-1))),2)}$

Hamming: $M \leq \frac{\var{field_sizes[0]^code_lengths[0]}}{\var{sum(map(comb(code_lengths[0],j),j,0..(errors_corrected[0])))}} \approx \var{precround((field_sizes[0]^code_lengths[0])/sum(map(comb(code_lengths[0],j),j,0..(errors_corrected[0]))),2)}$

Singleton: $M \leq \var{field_sizes[0]}^{\var{code_lengths[0]-min_distances[0]+1}} = \var{field_sizes[0]^(code_lengths[0]-min_distances[0]+1)}$

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet

This question is not used in any exams.

Loading...

Error

Upper and lower bounds for maximum number of codewords

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 10 years, 1 month ago

Christian Lawson-Perfect 10 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: `list`

Parts

Gap-fill

a)

Loading...

Error

Upper and lower bounds for maximum number of codewords

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 10 years, 1 month ago

Christian Lawson-Perfect 10 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: list

Parts

Gap-fill

a)

Generated value: `list`