Find the syndrome of a word in a linear code

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From users who are members of Content created by Newcastle University :

Newcastle University Mathematics and Statistics

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6 years, 7 months ago

From users who are not members of Content created by Newcastle University :

marquina carmona

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4 years, 9 months ago

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marquina carmona 4 years, 9 months ago

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Christian Lawson-Perfect 6 years, 7 months ago

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Newcastle University Mathematics and Statistics 6 years, 7 months ago

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Christian Lawson-Perfect 6 years, 7 months ago

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Question statement

$C$ is a {nary} linear code with generator matrix

$\mathrm{G} =\var{G}$

Give any introductory information the student needs.

			Name	Type	Generated Value

field_size	integer	2
nary	string	binary
dimension	integer	3
word_length	integer	8
dual_dimension	integer	5
A	list	List of 3 items
M	list	List of 3 items
G_words	list	List of 3 items
G	matrix	Matrix of size 3×8
G_dual	matrix	Matrix of size 5×8
word	codeword	codeword("01011011",2)
word_syndrome	codeword	codeword("10000",2)
word_is_in_code	boolean	false
word_is_in_code_explanation	string	$\operatorname{syn} w \neq \ma

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2,3,5)

JME function reference

Description

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

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Generated value: `integer`

→ Used by:

A
nary
word
word_is_in_code

Parts

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

Gap-fill

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Find the standard parity check matrix $\mathrm{H}$ for $C$.

$\mathrm{H} = $ Unnamed gap

Advice

First, row-reduce $\mathrm{G}$ to obtain a standard generator matrix $\mathrm{M}$ for the code

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

This is in the form $(\mathrm{I}_{\var{dimension}}|\mathrm{A})$. The standard parity check matrix is then $(-\mathrm{A^T}|\mathrm{I}_{\var{word_length-dimension}})$, i.e.

\[ \mathrm{H} = \var{G_dual} \]

If $w = \var{word}$ then $\operatorname{syn} w = w \mathrm{H^T} = \var{word_syndrome}$.

{word_is_in_code_explanation}

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Part	Test	Passed?
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Matrix entry
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Gap-fill
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Match text pattern
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Find the syndrome of a word in a linear code

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

marquina carmona 4 years, 9 months ago

Christian Lawson-Perfect 6 years, 7 months ago

Newcastle University Mathematics and Statistics 6 years, 7 months ago

Christian Lawson-Perfect 6 years, 7 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

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Error

Find the syndrome of a word in a linear code

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

marquina carmona 4 years, 9 months ago

Christian Lawson-Perfect 6 years, 7 months ago

Newcastle University Mathematics and Statistics 6 years, 7 months ago

Christian Lawson-Perfect 6 years, 7 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`