6. Let be the binary repetition code with parameter [n, k] = [12, 4].
a) Give a generator matrix for that is in standard form. What does standard form means?
b) Give a parity check matrix for that is in standard form.
c) What is the minimum distance of ?
d) What is the minimum distance of ?
e) What is the dimension of ?
f) What is the dimension of ?
g) How many codewords are in ? List them.
h) How many codewords are in ? List them

6. Let be the binary repetition code with parameter [n, k] = [12, 4].
a) Give a generator matrix for that is in standard form. What does standard form means?
b) Give a parity check matrix for that is in standard form.
c) What is the minimum distance of ?
d) What is the minimum distance of ?
e) What is the dimension of ?
f) What is the dimension of ?
g) How many codewords are in ? List them.
h) How many codewords are in ? List them

Solution 59

Let £ be the binary repetition code => n = 12, k = 4, m = n - k = 12 - 4 = 8

a) Standard form means the form in which the generator matrix can be expressed as

G = [ Ik P ]

Where Ik is an identity matrix of order ( k x k) and P is a order ( k x m)

Generator matrix is standard form is given by

G = [ 1 ...

Solution Summary

This provides several questions about binary repetition code including generator and parity check matrices, distance, dimension, and codewords.

The weight of w(x) of a vector x in (F_q )^n is defined to be the number of nonzero entries of x.
Prove that, in a binary linear code, either all the codewords have even weight or exactly half even weight and half odd weight.

Convert any binary number found in register AX to the equivalent ASCII code for each hexadecimal nibble in AX. Place the ASCII code for the hexadecimal number in AX starting at offset location 0000.
Example:
AX = 3030 ;ASCII for 00
[0800] = 41h after conversion (ASCII A)
Note:
Please write in assembly language progr

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Group Code of an Encoding Function
Show that the (3, 7) encoding function e: B^3 --> B^7 defined by
e(000) = 0000000

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Note: To find all cyclic codes of length n, find all ideals in B[x]/x^2+1
Note: If 1 is an Ideal (I) then R = I.
Example:
n=2
R=B[x]/x^2+1, x^2=1
R={o,1,x,1+x}
Ideals <0> = 0
<1> = R
x = (0, x, x^2...)
= (1,...

PROBLEM: For what N is it possible to list all the positive integers less than N in a Gray code, i.e., in such a way that successive numbers differ in exactly one position when the numbers are represented in binary form? For example, we can do so for N = 4 since the numbers 1, 2, 3 can be listed as 01, 11, 10 in binary form, whe

See the attached file.
Consider the (3,8) encoding function e:B^3 --> B^8 defined by
e(000) = 00000000
e(001) = 10111000
e(010) = 00101101
e(011) = 10010101

Consider a binary symmetric channel with error probability p = 0.1. Assume un-coded words of length n bits are transmitted over this channel. What is the word error probability? Suppose now a single error correcting code with length n is used. What is the word error probability?

1.- Find a parity check matrix for the [12,4] repetitioncode.
Can you expalin what does mean repetitioncode and the parity check matrix?
2.- Show that the complement of each codeword in the [12,4] repetitioncode is again a codeword.
Note: do not do this problem by checking all the sums of codewords.
Please Can yo