Word-length, dimension and minimum distance of Hamming codes

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Question statement

Write down the word-length, dimension and minimum distance of $C$ , where

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			Name	Type	Generated Value

binary_hamming_rows	integer	7
binary_word_length	number	127
binary_dimension	number	120
hamming_prime	integer	3
hamming_rows	integer	3
hamming_word_length	number	13
hamming_dimension	number	10
parity_hamming_rows	integer	9
parity_hamming_word_length	number	512
parity_hamming_dimension	number	502
simplex_prime	integer	5
simplex_rows	integer	6
simplex_word_length	number	3906
simplex_dimension	integer	6
simplex_minimum_distance	number	3125

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$C = \mathrm{Ham}(\var{binary_hamming_rows})$

Word-length: Gap 0.

Dimension: Gap 1.

Minimum distance: Gap 2.

Advice

a)

$\operatorname{Ham}(\var{binary_hamming_rows})$ has word-length $\simplify[]{2^{binary_hamming_rows}-1 = {binary_word_length}}$, dimension $\simplify[]{{binary_word_length}-{binary_hamming_rows} = {binary_dimension}}$ and minimum distance $3$.

b)

$\operatorname{Ham}_{\var{hamming_prime}}(\var{hamming_rows})$ has word-length $\simplify[]{({hamming_prime}^{hamming_rows}-1)/({hamming_prime}-1) = {hamming_word_length}}$, dimension $\simplify[]{{hamming_word_length}-{hamming_rows} = {hamming_dimension}}$ and minimum distance $3$.

c)

$\operatorname{Ham}^+(\var{parity_hamming_rows})$ has word-length $\simplify[]{2^{parity_hamming_rows} = {parity_hamming_word_length}}$, dimension $\simplify[]{{parity_hamming_word_length}-1-{parity_hamming_rows} = {parity_hamming_dimension}}$ and minimum distance $4$.

d)

$\operatorname{Sim}_{\var{simplex_prime}}(\var{simplex_rows})$ has word-length $\simplify[]{({simplex_prime}^{simplex_rows}-1)/({simplex_prime}-1) = {simplex_word_length}}$, dimension $\var{simplex_rows}$ and minimum distance $\simplify[]{{simplex_prime}^({simplex_rows}-1)={simplex_minimum_distance}}$.

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Word-length, dimension and minimum distance of Hamming codes

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Taxonomy: mathcentre

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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

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Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

d)

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Word-length, dimension and minimum distance of Hamming codes

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

Gap-fill

a)

b)

c)

d)

Generated value: `integer`