Generator matrix for Simplex code

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Newcastle University Mathematics and Statistics

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Christian Lawson-Perfect 4 years, 10 months ago

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Christian Lawson-Perfect 5 years, 11 months ago

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Newcastle University Mathematics and Statistics 6 years, 8 months ago

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Question statement

Give any introductory information the student needs.

			Name	Type	Generated Value

p	integer	2
rows	integer	3
word_length	number	7

			Name	Type	Generated Value

			pcm	list	List of 3 items
			gen_matrix	list	List of 4 items

			Name	Type	Generated Value

word	codeword	codeword("0110011",2)
syndrome	codeword	codeword("0000",2)
word_belongs	boolean	true

			Name	Type	Generated Value

swap	permutation	perm("(1,3,4)")
iswap	permutation	perm("(1,4,3)")
permuted_pcm	matrix	Matrix of size 3×7

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2,3)

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) Matrix entry
Part c) Gap-fill
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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

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Minimum number of columns

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Margin of error allowed in each cell

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Pre-filled cells

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​

Advice

a)

The lexicographic parity check matrix for $\operatorname{Ham}_{\var{p}}(\var{rows})$ is

\[ \mathrm{H} = \var{codeword_matrix(pcm)} \]

b)

Performing the column permutation $\var{perm_arrows(swap)}$ on $\mathrm{H}$ gives

\[ H^{\ast} = \var{permuted_pcm} \]

which we can invert (via $(\mathrm{I}|\mathrm{A}) \mapsto (-\mathrm{A^T}|\mathrm{I})$), and then perform the reverse permutation, $\var{perm_arrows(iswap)}$ to get

\[ \mathrm{G} = \var{codeword_matrix(gen_matrix)} \]

This is a generator matrix for $\operatorname{Ham}_{\var{p}}(\var{rows})$. It is also a parity-check matrix for its dual code $\operatorname{Sim}_{\var{p}}(\var{rows})$.

c)

Using $\mathrm{G}$, we can check whether $w = \var{word}$ is a codeword of $\operatorname{Sim}_{\var{p}}(\var{rows})$ by calculating its syndrome $w \mathrm{G^T}$: this is $\var{syndrome}$. The syndrome is not $\mathbf{0}$, so $w$ is not a codeword. The syndrome is $\mathbf{0}$, so $w$ is a codeword.

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Christian Lawson-Perfect 4 years, 10 months ago

Christian Lawson-Perfect 5 years, 11 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

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Warning

Generated value: `integer`

Parts

Matrix entry

Advanced settings

a)

b)

c)

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Generator matrix for Simplex code

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 4 years, 10 months ago

Christian Lawson-Perfect 5 years, 11 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`