Find a generator matrix for a code and decide if it is standard - binary

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Christian Lawson-Perfect 6 years, 7 months ago

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Question statement

Let $C$ be the {nary} code generated by the following set of codewords.

$\var{join_words(codewords)}$

Give any introductory information the student needs.

			Name	Type	Generated Value

word_length	integer	5
field_size	integer	2
num_words	integer	3
nary	string	binary

			Name	Type	Generated Value

num_skips	integer	0
data_vectors	list	List of 3 items
independent_combos	list	List of 3 items
extra_combos	list	List of 2 items
skips	list	[ ]
rest	matrix	Matrix of size 3×5
g_standard_bit	matrix	Matrix of size 3×5

			Name	Type	Generated Value

initial_g	matrix	Matrix of size 3×5
shuffled_g	matrix	Matrix of size 3×5
combos	list	List of 5 items
codewords	list	List of 5 items
g	list	List of 3 items
is_standard	boolean	true
encoded_data_vectors	list	List of 3 items

			Name	Type	Generated Value

			to_decode	codeword	codeword("10101",2)
			decoded_word	codeword	codeword("101",2)

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
5

JME function reference

Description

The length of each codeword.

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

Can an exam override the value of this variable?

Generated value: `integer`

→ Used by:

g_standard_bit
num_skips
rest

Parts

Part a) Matrix entry
Part b) Choose one from a list
Part c) Gap-fill
1. Gap Gap 0. Match text pattern
  Add
2. Gap Gap 1. Match text pattern
  Add
3. Gap Gap 2. Match text pattern
  Add
Part d) Match text pattern

Matrix entry

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

If there is a standard generator matrix for this code, enter it below. Otherwise, write down a generator matrix with the fewest possible rows, in reduced row-echelon form.

Marks

Correct answer

xxxxxxxxxx
 
codeword_matrix(g)

Show correct answer on reveal?

Show score feedback icon?

Advanced settings

Number of rows

Number of columns

Allow student to change size of matrix?

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

Gain marks for each correct cell?

Precision restriction

Allow the student to enter fractions?

Display numbers in the correct answer as fractions?

Pre-filled cells

​x
 
​

Advice

ca)

Write down the matrix whose rows are the given codewords:

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

Now reduce this matrix to row echelon form. That means the first non-zero entry in each row is further right than in the row above it and that entry is a 1 and there are only zeros above and below that entry. Remove any rows made up of all zeros.

From the matrix above, we obtain:

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

b)

A generator matrix for a code is a standard generator matrix if it is of the form $(\mathrm{I}_{k}|\mathrm{A})$, that is, the $k \times k$ identity matrix attached to an arbitrary matrix $\mathrm{A}$.

If a code has a standard generator matrix, the row reduced echelon form will be it.

$\mathrm{G}$ is of the required form, so it is a standard generator matrix.$\mathrm{G}$ is not of the required form, so it is not a standard generator matrix.

c)

To encode a data vector $\mathbf{v}$, multiply it on the right by the generator matrix for the code.

So we have:

\begin{align}
\simplify[rowvector]{{vector(data_vectors[0])}*{codeword_matrix(g)}} &= \var[rowvector]{vector(encoded_data_vectors[0])} \\[0.5em]
\simplify[rowvector]{{vector(data_vectors[1])}*{codeword_matrix(g)}} &= \var[rowvector]{vector(encoded_data_vectors[1])} \\[0.5em]
\simplify[rowvector]{{vector(data_vectors[2])}*{codeword_matrix(g)}} &= \var[rowvector]{vector(encoded_data_vectors[2])} \\[0.5em]
\end{align}

That is,

\begin{align}
\var{data_vectors[0]} &\mapsto \var{encoded_data_vectors[0]} \\[1em]
\var{data_vectors[1]} &\mapsto \var{encoded_data_vectors[1]} \\[1em]
\var{data_vectors[2]} &\mapsto \var{encoded_data_vectors[2]} \\[1em]
\end{align}

d)

We are looking for a data vector of length $\var{len(g)}$ which is encoded as $c = \var{to_decode}$.

Note that, since $\mathrm{G}$ is in reduced echelon form, the leading non-zero entry in each row is $1$, so the elements of the data vector are the digits of $c$ corresponding to each of the columns of $G$ containing a leading non-zero entry. The leading $1$s in $\mathrm{G}$ are in columns {join(leading_columns(g),', ')}. The corresponding digits of $c$ give the data vector $\var{decoded_word}$.

We can confirm that this vector is encoded as $c$:

\[ \simplify[rowvector]{ {vector(decoded_word)}*{codeword_matrix(g)} } = \var[rowvector]{vector(to_decode)} \]

Give a worked solution to the whole question.

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Find a generator matrix for a code and decide if it is standard - binary

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

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Comment

Checkpoint description

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

Matrix entry

Advanced settings

ca)

b)

c)

d)

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Error

Find a generator matrix for a code and decide if it is standard - binary

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

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

Matrix entry

Advanced settings

ca)

b)

c)

d)

Generated value: `integer`