Hamming code notes

Error correcting coding - Hamming code

Hamming code: can detect double- bit errors and correct single- bit errors.

work process

Confirm the number of check code digits r
Determine the location of the checksum and data
Find the value of the check code
Detect and correct errors

01 Confirm the number of check code digits r

$bit\text{Hamming's inequality:} 2^r \geq k + r + 1 \\ k is the information bit, r is redundant information bits$

Principle :

There are a total of r bits of redundant information bits $2^r$ combinations, except for 1 whose identification information is correct, there are $2^r-1$ combination to identify $2^r - 1$ position error. Then there should be the number of identification error combinations >= the total number of transmitted bits, that is, $2^r - 1 \geq k + r$

02 Determine the location of the check code and data

Assume data $D = 101101$

Then according to Hamming's inequality, $r = 4$

Let the check digits from left to right be $P_1, P_2, P_3, P_4$ , the data from left to right are $D_1, D_2, D_3, D_4, D_5, D_6$

The check digits are placed in several powers of 2, and the data bits are filled in the empty bits from left to right.

Location	1	2	3	4	5	6	7	8	9	10
code	$P_1$	$P_2$	$D_1$	$P_3$	$D_2$	$D_3$	$D_4$	$P_4$	$D_5$	$D_6$
actual value			1		0	1	1		0	1

03 Find the value of the check code

Each check digit is filled in a power of 2 position, so its "binary representation of position" has only one 1, which is set in the r-th position from the right. This check digit verifies the data whose r-th bit from the right of the "position binary representation" is 1.

For example: $P_1$ In the positional binary representation, 1 is in the first digit from the right, then it verifies all data whose first digit from the right in the "positional binary representation" is 1, that is, the 3rd, 5th, 7th, 9th... 3, 5, 7, $\ldots$ bits, $P_2$ Check the $\ldots$ bits, $P_3$ Check the $\ldots$ bits, $P_4$ Check the $\ldots$ bit.

The parity bit should make the data in all parity positions XOR with itself to be 0 (can be understood as even parity).
$\begin{aligned} & P_1 \oplus D_1 \oplus D_2 \oplus D_4 \oplus D5 = 0 & \implies & P_1 \oplus 1 \oplus 0 \oplus 1 \oplus 0 = 0 & \implies P_1 = 0 \\ & P_2 \oplus D_1 \oplus D_3 \oplus D_4 \oplus D6 = 0 & \implies & P_2 \oplus 1 \oplus 1 \oplus 1 \ oplus 1 = 0 & \implies P_2 = 0 \\ & P_3 \oplus D_2 \oplus D_3 \oplus D_4 = 0 & \implies & P_3 \oplus 0 \oplus 1 \oplus 1 = 0 & \implies P_3 = 0 \\ & P_4 \oplus D_5 \oplus D_6 = 0 & \implies & P_4 \oplus 0 \oplus$

Fill in the resulting check digit value

Location	1	2	3	4	5	6	7	8	9	10
binary representation	0001	0010	0011	0100	0101	0110	0111	1000	1001	1010
code	$P_1$	$P_2$	$D_1$	$P_3$	$D_2$	$D_3$	$D_4$	$P_4$	$D_5$	$D_6$
actual value	0	0	1	0	0	1	1	1	0	1

That is, the transmitted data is 0010 0111 01

04 Error detection and correction

Assume that the 5th bit during transmission is $D_2$ An error occurred, the result received was 0010 1 111 01

Perform XOR calculation according to the check value method
$\ begin{aligned} & P_1 \oplus D_1 \oplus D_2 \oplus D_4 \oplus D5 & = & 0 \oplus 1 \oplus 1 \oplus 1 \oplus 0 & = 1 & \\ & P_2 \oplus D_1 \oplus D_3 \oplus D_4 \oplus D6 & = & 0 \oplus 1 \oplus 1 \oplus 1 \oplus 1 & = 0 \\ & P_3 \oplus D_2 \oplus D_3 \oplus D_4 & = & 0 \oplus 1 \oplus 1 \oplus 1 & = 1 \\ & P_4 \oplus D_5\oplus D_6 & = & 1 \oplus 0 \oplus 1 & = 0 \end{aligned}$
The result of sorting from bottom to top is 0101, which corresponds to decimal 5, that is, the 5th bit is wrong, just change it from 1 to 0.

Note: Both single-bit and double-bit errors can be detected, but only single-bit errors can be corrected. If a double-bit error occurs, the Hamming code check result can only indicate an error, that is, the check result is not all 0, but the corresponding position is not necessarily wrong. In this case, the Hamming code defaults to a single-bit error and is corrected. , there will be error correction and error problems.

Principle : When a certain check bit error detection XOR output is 1, it means that the position of the "position binary representation" of the error bit must be 1. If the XOR output is 0, it means that the position of the "position binary representation" of the error bit must be 1. The position is 0. Then the check results sorted from bottom to top exactly represent the "position binary representation" of the error bit.

My personal understanding. If there are any mistakes or you have a better understanding, please correct me.

The above are personal study notes, study video link B Station-Wangdao Postgraduate Entrance Examination-Computer Network