Construct a Slepian array and find coset leaders

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Christian Lawson-Perfect	said	Should not be used	1 year, 11 months ago
Newcastle University Mathematics and Statistics	said	Ready to use	6 years, 7 months ago

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Christian Lawson-Perfect commented 1 year, 11 months ago

Marking as "should not be used" because it has a custom JS marking script which doesn't work.

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Christian Lawson-Perfect 1 year, 11 months ago

Gave some feedback: Should not be used

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Christian Lawson-Perfect 6 years, 7 months ago

Published this.

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Newcastle University Mathematics and Statistics 6 years, 7 months ago

Gave some feedback: Ready to use

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Christian Lawson-Perfect 6 years, 7 months ago

Created this.

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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	6
dual_dimension	integer	3
A	list	List of 3 items
M	list	List of 3 items
g_words	list	List of 3 items
g	matrix	Matrix of size 3×6
pcm	list	List of 3 items
G_dual	matrix	Matrix of size 3×6
slep	list	Nested 8×8 list
coset_leaders	list	List of 8 items
syndromes	list	List of 8 items
C_words	list	List of 8 items
C	code	code([codeword("000000",2), codeword("001110",2), codeword("010001",2), codeword("011111",2), codeword("100000",2), codeword("101110",2), codeword("110001",2), codeword("111111",2)])
word	codeword	codeword("100111",2)
word_index	number	7
word_coset	list	List of 8 items
transmitted_word	codeword	codeword("110001",2)
received_word	codeword	codeword("111001",2)
q	codeword	codeword("001000",2)
unique_coset_leader_indices	list	[ 2, 3, 4 ]

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Description

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Generated value: `integer`

→ Used by:

A
nary
transmitted_word
word

Parts

Part a) Gap-fill
1. Gap Matrix entry
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Part b) Match text pattern
Part c) Match text pattern
Part d) Match text pattern
Part e) Match text pattern
Part f) Match text pattern

Gap-fill

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

$\mathrm{H} = $ Unnamed gap

Advice

a)

Since the reduced row echelon form of $\mathrm{G}$ is in standard form, we can write down the standard parity check matrix,

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

b)

The code consists of all possible combinations of the rows of $\mathrm{G}$. There are 3 words in the basis, so $2^3 = 8$ words in the whole code $C$.

c)

Now we need to write out a Slepian array for $C$. Each row of the Slepian array corresponds to a coset of $C$.

There are various solutions for the Slepian array, depending on our choice of coset leaders. A natural solution is

{table(map(map(inline_latex(y),y,x),x,slepian_array(g_words)),[])}

This is constructed by first writing out all the codewords of $C$ as the first row and then, to obtain each subsequent row, finding the word of minimum weight which isn't already in the array, and adding it to all the words of $C$.

The coset containing $\var{word}$ is the row of the Slepian array in which it occurs. That is,

$\var{latex(join(map(latex(x),x,word_coset),' \\quad '))}$

d)

The coset leader of a coset is an element of minimum weight. There could be more than one choice of coset leader for a coset.

The first column of the Slepian array gives a set of coset leaders.

e)

The syndrome of a word $w$ with respect to a parity matrix $\mathrm{H}$ is $w\mathrm{H^T}$.

For example, the syndrome of $\var{coset_leaders[1]}$ is

\[ \operatorname{syn} (\var{coset_leaders[1]}) = (\var{coset_leaders[1]}) \var{transpose(G_dual)} = \var{syndromes[1]} \]

f)

One way of decoding the given word is to find it in the Slepian array, and correct it to the corresponding word in the same column of the first row.

Another way is to compute its syndrome, and subtract the coset leader with the corresponding syndrome.

We can find that $\operatorname{syn}(\var{received_word}) = \var{syndrome(received_word,pcm)}$. This corresponds to the coset with leader $\var{received_word-transmitted_word}$, so the received word should be corrected to $\var{received_word} - \var{received_word-transmitted_word} = \var{transmitted_word}$.

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Construct a Slepian array and find coset leaders

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Taxonomy: mathcentre

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Checkpoint description

Christian Lawson-Perfect commented 1 year, 11 months ago

Christian Lawson-Perfect 1 year, 11 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

a)

b)

c)

d)

e)

f)

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Error

Construct a Slepian array and find coset leaders

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect commented 1 year, 11 months ago

Christian Lawson-Perfect 1 year, 11 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

a)

b)

c)

d)

e)

f)

Generated value: `integer`