Hamming distances

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Question statement

A set of words in $\mathbb{Z}_2^5$ is described below. You will be asked to encode them, calculate the minimum distance between code words, and describe how many errors can be detected or corrected.

Encoding is described as follows:

First, repeat the word. So if $w = 01101$ , then we would get $0110101101$ .
Add a parity bit to the end based on the original word. Since $\mathrm{wt}(w) = 3$ , then the parity bit is $1$ .
So $E(w) = 01101011011$ .

The set of words is $W = \{ \var{binstring1}, \var{binstring2}, \var{binstring3}, \var{binstring4}\}$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

binstring1	string	10010
binstring2	string	10100
binstring3	string	11010
binstring4	string	11110
bins1	list	List of 5 items
bins2	list	List of 5 items
bins3	list	List of 5 items
bins4	list	List of 5 items
err12	number	4
err13	number	3
err14	number	4
err23	number	7
err24	number	4
err34	number	3
minerror	number	3
errors	list	[ 4, 3, 4, 7, 4, 3 ]
detect	number	2
correct	integer	1
code1	list	List of 11 items
code2	list	List of 11 items
code3	list	List of 11 items
code4	list	List of 11 items
par12	number	0
par22	number	0
par32	number	1
par42	number	0
codes1	list	List of 11 items
codes2	list	List of 11 items
codes3	list	List of 11 items
codes4	list	List of 11 items
codeword1	string	10010100100
codeword2	string	10100101000
codeword3	string	11010110101
codeword4	string	11110111100

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10010

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The minimum distance is the smallest non-zero value of $d(x,y)$ where $x$ , $y \in C$ . Call this $m$ .

If the minimum distance of the code is $k$ , then the code can detect up to $m-1$ errors.

If the minimum distance of the code is $k$ , then the code can correct up to $\lfloor \frac{m-1}{2} \rfloor$ .

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Alexander Corner 3 years, 2 months ago

Alexander Corner 3 years, 2 months ago

Alexander Corner 3 years, 2 months ago

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Hamming distances

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

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Comment

Checkpoint description

Alexander Corner 3 years, 2 months ago

Alexander Corner 3 years, 2 months ago

Alexander Corner 3 years, 2 months ago

Error in variable testing condition

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Generated value: string

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Generated value: `string`