A 3-bit redundancy-check code (odd parity for columns, even parity for rows) has

Question

A 3-bit redundancy-check code (odd parity for columns, even parity for rows) has been used to transmit data. This means that the original data had words of 2 bits each. The following sequence is received (notice that one error occured):

0 0 0 1 0 1 0 1 0 0 0 0 1 0 1

Remember that the very last bit should check for even parity on the last word (row), not for odd parity on the last column; you do not need to worry about the check for odd parity on the last column. For example, if you have the two 3-bit words 000 and 110, you know that the first word is correct (even parity), the second word is correct (even parity), the first column is correct (odd parity), the second column is correct (odd parity). Checking the last column would tell you something is not correct (it is of even parity), when obviously everything is fine! Checking the last column is useless and could even be misleading. The parity of the last column also depends, by construction, on the number of words in the data.
a) How many of the bits in the above sequence are redundant bits (i.e. they were added as part of the redundancy-check code before transmission), and how many bits are for the data only?

Redundant Bits:

Data Bits:


b) State in which word the error occured (1 for leftmost - 5 for rightmost)


1
2
3
4
5

c) State in which bit position (1 for left, 2 for middle, 3 for right) in that word:


1
2
3 (parity bit)

d) Give the original data sequence that was sent (correct the error, and eliminate the added bits used for error detection/correction)

Explanation / Answer

as there are 15 bits and the word length is of 3 bits (with parity bit) so we can write the transmitted data in the form of 5 rows and 3 columns

so the matrix will be

          0    0    0

x=      1    0     1

          0    1    0

          0     0    0

           1    0    1

in this matrix we will check for evn parity for rows and odd parity for columns.

while looking at rows we can find that in third row the data is 0 1 0. as this data contains only one 1. so for the even parity it should be 1. i.e the data would be   0 1 1. from here we found the error is in third row or in the 3rd word transmitted but not the exact position of which bit is corrupted. Now to look at what position the error is we have to chech for the odd parity in the columns.

by looking at the 1st column we found that the column is 0 1 0   0   1. considering that the last bit can be viewed as odd parity check bit. here it is 1 though there are one 1. so it tells that the 1st bit is error.

a)   as the last row is for odd parity check it is a redundacy data. In addition to this, there are remaing bits of 15-3=12 bits.

As the word lenght is 3 so there are 12/3 = 4 words. each word has a parity check bit. so there are 4 even parity check bits.

so total parity check bits are 4 (even parity) + 3 (odd parity) = 7 bits

There are 7 bits of redundacy data and 8 bits of original data only.

b)

As i stated above the error occured in the 3rd row . that the 3rd word is erroeneous.

hence the answer is 3.

c)

As i have mentioned above the first column is erroneous so the first bit in 3rd word is error. the received data should be   1 1   0.

hence answer is   1.

d)

the correct data sent was

00101100

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