# Orthogonality of a Generator Matrix of a Binary Block Code

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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?

https://brainmass.com/math/matrices/orthogonality-generator-matrix-binary-block-code-98055

#### Solution Preview

Please see the attached file.

Orthogonality

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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

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For part b, you need to understand the concept

of Generator and Linear ...

#### Solution Summary

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