Lexicographic parity check matrix - ternary, 5-ary or 7-ary

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Christian Lawson-Perfect 6 years, 8 months ago

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Newcastle University Mathematics and Statistics 6 years, 8 months ago

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Question statement

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			Name	Type	Generated Value

p	integer	5
rows	integer	2
pcm	list	List of 2 items
word_length	number	6
num_words	integer	2
gen_matrix	list	List of 4 items
words	list	List of 2 items
error_positions	list	[ 5, 2 ]
received_words	list	List of 2 items
syndromes	list	List of 2 items
multiply_error	list	[ 1, 4 ]
columns	list	List of 6 items

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(3,5,7)

JME function reference

Description

Size of the field in which the code lives

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

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Generated value: `integer`

This variable doesn't seem to be used anywhere.

Parts

Part a) Matrix entry
Part b) Gap-fill
1. Gap Match text pattern
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Part c) Gap-fill
1. Gap Match text pattern
  Add

Matrix entry

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

Write down the lexicographic parity check matrix for $\operatorname{Ham}_{\var{p}}(\var{rows})$ .

Marks

Correct answer

xxxxxxxxxx
 
codeword_matrix(pcm)

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Number of rows

Number of columns

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Minimum number of rows

Maximum number of rows

(No limit)

Minimum number of columns

Maximum number of columns

(No limit)

Margin of error allowed in each cell

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Pre-filled cells

​x
 
​

Advice

a)

The lexicographic parity check matrix for $\operatorname{Ham}_{\var{p}}(\var{rows})$ has the following properties:

Property 1: The first non-zero entry of each column is $1$.
Property 2: The columns are ordered from smallest to largest.

The unique parity-check matrix for $\operatorname{Ham}_{\var{p}}(\var{rows})$ with this property is

\[ \var{codeword_matrix(pcm)} \]

b)

The word $\var{received_words[0]}$ has syndrome

\[ (\var{received_words[0]}) \var{transpose(codeword_matrix(pcm))} = (\var{syndromes[0]}) \]

$(\var{syndromes[0]}) = \var{multiply_error[0]} \times (\var{columns[error_positions[0]]})$. Since $\var{columns[error_positions[0]]}$ is the {nth(error_positions[0]+1)} column of the parity check matrix, we subtract $\var{multiply_error[0]}$ from the {nth(error_positions[0]+1)} column to get $\var{words[0]}$.

c)

The word $\var{received_words[1]}$ has syndrome

\[ (\var{received_words[1]}) \var{transpose(codeword_matrix(pcm))} = (\var{syndromes[1]}) \]

$(\var{syndromes[1]}) = \var{multiply_error[1]} \times (\var{columns[error_positions[1]]})$. Since $\var{columns[error_positions[1]]}$ is the {nth(error_positions[1]+1)} column of the parity check matrix, we subtract $\var{multiply_error[1]}$ from the {nth(error_positions[1]+1)} column to get $\var{words[1]}$.

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Lexicographic parity check matrix - ternary, 5-ary or 7-ary

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Christian Lawson-Perfect 6 years, 8 months ago

Newcastle University Mathematics and Statistics 6 years, 8 months ago

Christian Lawson-Perfect 6 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Matrix entry

Advanced settings

a)

b)

c)

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Error

Lexicographic parity check matrix - ternary, 5-ary or 7-ary

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 6 years, 8 months ago

Newcastle University Mathematics and Statistics 6 years, 8 months ago

Christian Lawson-Perfect 6 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Matrix entry

Advanced settings

a)

b)

c)

Generated value: `integer`