An error control code uses three parity checks over 4 data digits to construct c
ID: 2249563 • Letter: A
Question
An error control code uses three parity checks over 4 data digits to construct codewords containing seven digits. The parity checks are computed as the Exclusive-Or sum of data digits 1, 2, 3; 2, 3, 4; 1, 3, 4, respectively. a) List the set of all possible codewords b) Develop a table that relates the pattern of checks and failures to the location of a single error that may be present anywhere in the word received from a transmission channel c) Show that the table in b) is only a function of the location of the error, not a function of the particular codeword on which the error occurred. d) Assume a codeword suffers errors in locations 2 and 6 during transmission. Show the result of applying the error locating scheme in b) e) Show that the above code can locate all single errors correctly but is unable to correct any error pattern of multiplicity higher than 1.
An error control code uses three parity checks over 4 data digits to construct codewords containing seven digits. The parity checks are computed as the Exclusive-Or sum of data digits 1, 2, 3; 2, 3, 4; 1, 3, 4, respectively. a) List the set of all possible codewords b) Develop a table that relates the pattern of checks and failures to the location of a single error that may be present anywhere in the word received from a transmission channel c) Show that the table in b) is only a function of the location of the error, not a function of the particular codeword on which the error occurred. d) Assume a codeword suffers errors in locations 2 and 6 during transmission. Show the result of applying the error locating scheme in b) e) Show that the above code can locate all single errors correctly but is unable to correct any error pattern of multiplicity higher than 1.
An error control code uses three parity checks over 4 data digits to construct codewords containing seven digits. The parity checks are computed as the Exclusive-Or sum of data digits 1, 2, 3; 2, 3, 4; 1, 3, 4, respectively. a) List the set of all possible codewords b) Develop a table that relates the pattern of checks and failures to the location of a single error that may be present anywhere in the word received from a transmission channel c) Show that the table in b) is only a function of the location of the error, not a function of the particular codeword on which the error occurred. d) Assume a codeword suffers errors in locations 2 and 6 during transmission. Show the result of applying the error locating scheme in b) e) Show that the above code can locate all single errors correctly but is unable to correct any error pattern of multiplicity higher than 1.
Explanation / Answer
a.
Step 1: Make a list of Values data digits can acquire in ascending order and find P1, P2 and P3 doing XOR
Following set of all possible codewords at the transmitter
b. Lets all check as C1, C2 and C3
C1 is XOR(D1,D2,D3,P1)
C2 is XOR(D2,D3,D3,P2)
C3 is XOR(D1,D3,D4,P3)
If there is no error the table appears as follows at reciever
Following is the table that relates check code to position of error. It is similar to Hamming code
c) We can prove this by example.
In first case lets take Cordword transmitted to be
and recieved word to be
When we apply table derived in b) we get
TRUE
C1 and C2 are true and from table in b,error is in D2.
Now if we the error is introduced at D2 in some other word. Say transmitter word is
And received word has error in D2 as follows
Still C1 and C2 have True and error is detected in D2 for different word.
Hence proved.
d)
Transmitted word be
It suffers errors in D2 and P2
It detects error in PC1
e) The example in d) proves that the code can not attend multiplicity of errors.
P3 P2 P1 D4 D3 D2 D1 FALSE FALSE FALSE 0 0 0 0 TRUE FALSE TRUE 0 0 0 1 FALSE TRUE TRUE 0 0 1 0 TRUE TRUE FALSE 0 0 1 1 TRUE TRUE TRUE 0 1 0 0 FALSE TRUE FALSE 0 1 0 1 TRUE FALSE FALSE 0 1 1 0 FALSE FALSE TRUE 0 1 1 1 TRUE TRUE FALSE 1 0 0 0 FALSE TRUE TRUE 1 0 0 1 TRUE FALSE TRUE 1 0 1 0 FALSE FALSE FALSE 1 0 1 1 FALSE FALSE TRUE 1 1 0 0 TRUE FALSE FALSE 1 1 0 1 FALSE TRUE FALSE 1 1 1 0 TRUE TRUE TRUE 1 1 1 1