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Find a parity check matrix for the [12,4] repetition code.

1.- Find a parity check matrix for the [12,4] repetition code.
Can you expalin what does mean repetition code and the parity check matrix?

2.- Show that the complement of each codeword in the [12,4] repetition code is again a codeword.
Note: do not do this problem by checking all the sums of codewords.

Please Can you do some sums computation to understand it?

Please can you explain them step by step in order to understand it.
If it is possible give some examples.
I am confused with this terminology.

Thank you so much.

Solution Preview

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Linear Block Codes

Block codes take a block of data, say k bits, adds redundancy to it and produces n bits, these are called code words. Diagrammatically:

(see attached file for diagram)

We say we have an [n,k] block code. So for example a [12,4] block code produces 12 bits for every 4 that enter the block encoder. These code words are then transmitted over a channel and may get corrupted, hopefully the redundancy we added in the encoder will help us work out the original message x. We could try weird and wonderful ways of encoding x to get y but let's start off simply. Let's assume we have a linear block code, i.e:

(see attached file for equation)

It's linear because matrix multiplication is linear. G is the generator matrix, it is a matrix that is k by n. Note that all our multiplications and additions are done modulo 2, such that y and x are binary vectors. It is only by convention ...

Solution Summary

This solution is comprised of a detailed explanation to find a parity check matrix for the [12,4] repetition code.