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Orthogonality of a Generator Matrix of a Binary Block Code

Please see the attached file for the fully formatted problems.
4. Let be the generator matrix of a binary block code .

a) Show that each row of G is orthogonal to itself and to each of the other rows of G.
b) Show that each codeword in is orthogonal to itself and to every other codeword in .
Note: Use part a) and the fact that each codeword is a linear combination of the rows of G.
c) What is the dimension of ? Explain
d) What is the dimension of ? Explain
e) Show that is a self-dual code, that is show that = .
Can you explain what does orthogonally means and self-dual code?

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Orthogonality
-------------
Originally, this concept means at right angles
or 90° or perpendicular. Then using vectors
we find that they are orthogonal if their
dot product is zero. For example,
vector a = [ 1 0 3 ]
vector b = [-3 7 1 ]
take the dot product,
a · b = 1×-3 + 0×7 + 3×1 = -3 + 0 + 3 = 0

Now we extend the concept to binary codes. But
you must remember the laws for binary codes are
slightly different.
0 + 0 = 0 0 × 0 = 0
0 + 1 = 1 0 × 1 = 0
1 + 0 = 1 1 × 0 = 0
1 + 1 = 0 1 × 1 = 1
Notice that 1+1=0
+ can be interpreted as + modulo 2: whenever
you have a 2, you always reset to 0
if you know electronics or computer science,
you can interpret this + as XOR (exclusive or)

× is similar to our ordinary ×
it can be interpreted in electronics or computer
science as AND.

Once you understand this, you can do part a).
[ 1 0 0 0 0 1 1 1 ]
G = [ 0 1 0 0 1 0 1 1 ]
[ 0 0 1 0 1 1 0 1 ]
[ 0 0 0 1 1 1 1 0 ]
Denote the rows by r1, r2, r3 and r4. We can
calculate all the dot products, for example

r1·r1
= 1×1 + 0×0 + 0×0 + 0×0 + 0×0 + 1×1 + 1×1 + 1×1
= 1 + 0 + 0 + 0 + 0 + 1 + 1 + 1
= 0
(remember, every time you get 1+1 it's reset to
zero. if you get 4 ones, an even number, the
final result is zero.)

r1·r2
= 1×0 + 0×1 + 0×0 + 0×0 + 0×1 + 1×0 + 1×1 + 1×1
= 0 + 0 + 0 + 0 + 0 + 0 + 1 + 1
= 0 (two ones, even number, so total is 0)

By the same type of calculation, you can check
that r1·r3=0, r1·r4=0, and also
r2·r1=0, r2·r2=0, r2·r3=0, r2·r4=0,
r3·r1=0, r3·r2=0, r3·r3=0, r3·r4=0,
r4·r1=0, r4·r2=0, r4·r3=0, r4·r4=0.

Generator and Linear Combination
--------------------------------
For part b, you need to understand the concept
of Generator and Linear ...

Solution Summary

The Orthogonality of a Generator Matrix of a Binary Block Code is investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

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