Codes predating Hamming

A number of simple error-detecting codes were used before Hamming codes, but none were as effective as Hamming codes in the same overhead of space.

General algorithm

The following general algorithm generates a single-error correcting (SEC) code for any number of bits. The main idea is to choose the error-correcting bits such that the index-XOR (the XOR of all the bit positions containing a 1) is 0. We use positions 1, 10, 100, etc. (in binary) as the error-correcting bits, which guarantees it is possible to set the error-correcting bits so that the index-XOR of the whole message is 0. If the receiver receives a string with index-XOR 0, they can conclude there were no corruptions, and otherwise, the index-XOR indicates the index of the corrupted bit.

An algorithm can be deduced from the following description:

1. Number the bits starting from 1: bit 1, 2, 3, 4, 5, 6, 7, etc.
2. Write the bit numbers in binary: 1, 10, 11, 100, 101, 110, 111, etc.
3. All bit positions that are powers of two (have a single 1 bit in the binary form of their position) are parity bits: 1, 2, 4, 8, etc. (1, 10, 100, 1000)
4. All other bit positions, with two or more 1 bits in the binary form of their position, are data bits.
5. Each data bit is included in a unique set of 2 or more parity bits, as determined by the binary form of its bit position.
   1. Parity bit 1 covers all bit positions which have the least significant bit set: bit 1 (the parity bit itself), 3, 5, 7, 9, etc.
   2. Parity bit 2 covers all bit positions which have the second least significant bit set: bits 2-3, 6-7, 10-11, etc.
   3. Parity bit 4 covers all bit positions which have the third least significant bit set: bits 4–7, 12–15, 20–23, etc.
   4. Parity bit 8 covers all bit positions which have the fourth least significant bit set: bits 8–15, 24–31, 40–47, etc.
   5. In general each parity bit covers all bits where the bitwise AND of the parity position and the bit position is non-zero.

If a byte of data to be encoded is 10011010, then the data word (using _ to represent the parity bits) would be __1_001_1010, and the code word is 011100101010.

The choice of the parity, even or odd, is irrelevant but the same choice must be used for both encoding and decoding.

This general rule can be shown visually:

Bit position		1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	…
Encoded data bits		p1	p2	d1	p4	d2	d3	d4	p8	d5	d6	d7	d8	d9	d10	d11	p16	d12	d13	d14	d15
Parity bit coverage	p1
	p2
	p4
	p8
	p16

Shown are only 20 encoded bits (5 parity, 15 data) but the pattern continues indefinitely. The key thing about Hamming Codes that can be seen from visual inspection is that any given bit is included in a unique set of parity bits. To check for errors, check all of the parity bits. The pattern of errors, called the error syndrome, identifies the bit in error. If all parity bits are correct, there is no error. Otherwise, the sum of the positions of the erroneous parity bits identifies the erroneous bit. For example, if the parity bits in positions 1, 2 and 8 indicate an error, then bit 1+2+8=11 is in error. If only one parity bit indicates an error, the parity bit itself is in error.

With m parity bits, bits from 1 up to ${\displaystyle 2^{m}-1}$ can be covered. After discounting the parity bits, ${\displaystyle 2^{m}-m-1}$ bits remain for use as data. As m varies, we get all the possible Hamming codes:

Parity bits	Total bits	Data bits	Name	Rate
2	3	1	Hamming(3,1) (Triple repetition code)	1/3 ≈ 0.333
3	7	4	Hamming(7,4)	4/7 ≈ 0.571
4	15	11	Hamming(15,11)	11/15 ≈ 0.733
5	31	26	Hamming(31,26)	26/31 ≈ 0.839
6	63	57	Hamming(63,57)	57/63 ≈ 0.905
7	127	120	Hamming(127,120)	120/127 ≈ 0.945
8	255	247	Hamming(255,247)	247/255 ≈ 0.969
9	511	502	Hamming(511,502)	502/511 ≈ 0.982
…
m	${\displaystyle n=2^{m}-1}$	${\displaystyle k=2^{m}-m-1}$	Hamming${\displaystyle (2^{m}-1,2^{m}-m-1)}$	${\displaystyle (2^{m}-m-1)/(2^{m}-1)}$

Construction of G and H

The matrix ${\displaystyle \mathbf {G}$ :={\begin{pmatrix}{\begin{array}{c|c}I_{k}&-A^{\text{T}}\\\end{array}}\end{pmatrix}}} is called a (canonical) generator matrix of a linear (n,k) code,

and ${\displaystyle \mathbf {H}$ :={\begin{pmatrix}{\begin{array}{c|c}A&I_{n-k}\\\end{array}}\end{pmatrix}}} is called a parity-check matrix.

This is the construction of G and H in standard (or systematic) form. Regardless of form, G and H for linear block codes must satisfy

${\displaystyle \mathbf {H} \,\mathbf {G} ^{\text{T}}=\mathbf {0} }$, an all-zeros matrix.^[6]

Since [7, 4, 3] = [n, k, d] = [2^m − 1, 2^m − 1 − m, 3]. The parity-check matrix H of a Hamming code is constructed by listing all columns of length m that are pair-wise independent.

Thus H is a matrix whose left side is all of the nonzero n-tuples where order of the n-tuples in the columns of matrix does not matter. The right hand side is just the (n − k)-identity matrix.

So G can be obtained from H by taking the transpose of the left hand side of H with the identity k-identity matrix on the left hand side of G.

The code generator matrix ${\displaystyle \mathbf {G} }$ and the parity-check matrix ${\displaystyle \mathbf {H} }$ are:

${\displaystyle \mathbf {G}$ :={\begin{pmatrix}1&0&0&0&1&1&0\\0&1&0&0&1&0&1\\0&0&1&0&0&1&1\\0&0&0&1&1&1&1\end{pmatrix}}_{4,7}}

and

${\displaystyle \mathbf {H}$ :={\begin{pmatrix}1&1&0&1&1&0&0\\1&0&1&1&0&1&0\\0&1&1&1&0&0&1\end{pmatrix}}_{3,7}.}

Finally, these matrices can be mutated into equivalent non-systematic codes by the following operations:^[6]

• Column permutations (swapping columns)
• Elementary row operations (replacing a row with a linear combination of rows)

Encoding

Example

From the above matrix we have 2^k = 2⁴ = 16 codewords. Let ${\displaystyle {\vec {a}}}$ be a row vector of binary data bits, ${\displaystyle {\vec {a}}=[a_{1},a_{2},a_{3},a_{4}],\quad a_{i}\in \{0,1\}}$. The codeword ${\displaystyle {\vec {x}}}$ for any of the 16 possible data vectors ${\displaystyle {\vec {a}}}$ is given by the standard matrix product ${\displaystyle {\vec {x}}={\vec {a}}G}$ where the summing operation is done modulo-2.

For example, let ${\displaystyle {\vec {a}}=[1,0,1,1]}$. Using the generator matrix ${\displaystyle G}$ from above, we have (after applying modulo 2, to the sum),

${\displaystyle {\vec {x}}={\vec {a}}G={\begin{pmatrix}1&0&1&1\end{pmatrix}}{\begin{pmatrix}1&0&0&0&1&1&0\\0&1&0&0&1&0&1\\0&0&1&0&0&1&1\\0&0&0&1&1&1&1\\\end{pmatrix}}={\begin{pmatrix}1&0&1&1&2&3&2\end{pmatrix}}={\begin{pmatrix}1&0&1&1&0&1&0\end{pmatrix}}}$

[8,4] Hamming code with an additional parity bit

The same [7,4] example from above with an extra parity bit. This diagram is not meant to correspond to the matrix H for this example.

The [7,4] Hamming code can easily be extended to an [8,4] code by adding an extra parity bit on top of the (7,4) encoded word (see Hamming(7,4)). This can be summed up with the revised matrices:

${\displaystyle \mathbf {G}$ :={\begin{pmatrix}1&1&1&0&0&0&0&1\\1&0&0&1&1&0&0&1\\0&1&0&1&0&1&0&1\\1&1&0&1&0&0&1&0\end{pmatrix}}_{4,8}}

and

${\displaystyle \mathbf {H}$ :={\begin{pmatrix}1&0&1&0&1&0&1&0\\0&1&1&0&0&1&1&0\\0&0&0&1&1&1&1&0\\1&1&1&1&1&1&1&1\end{pmatrix}}_{4,8}.}

Note that H is not in standard form. To obtain G, elementary row operations can be used to obtain an equivalent matrix to H in systematic form:

${\displaystyle \mathbf {H} =\left({\begin{array}{cccc|cccc}0&1&1&1&1&0&0&0\\1&0&1&1&0&1&0&0\\1&1&0&1&0&0&1&0\\1&1&1&0&0&0&0&1\end{array}}\right)_{4,8}.}$

For example, the first row in this matrix is the sum of the second and third rows of H in non-systematic form. Using the systematic construction for Hamming codes from above, the matrix A is apparent and the systematic form of G is written as

${\displaystyle \mathbf {G} =\left({\begin{array}{cccc|cccc}1&0&0&0&0&1&1&1\\0&1&0&0&1&0&1&1\\0&0&1&0&1&1&0&1\\0&0&0&1&1&1&1&0\end{array}}\right)_{4,8}.}$

The non-systematic form of G can be row reduced (using elementary row operations) to match this matrix.

The addition of the fourth row effectively computes the sum of all the codeword bits (data and parity) as the fourth parity bit.

For example, 1011 is encoded (using the non-systematic form of G at the start of this section) into 01100110 where blue digits are data; red digits are parity bits from the [7,4] Hamming code; and the green digit is the parity bit added by the [8,4] code. The green digit makes the parity of the [7,4] codewords even.

Finally, it can be shown that the minimum distance has increased from 3, in the [7,4] code, to 4 in the [8,4] code. Therefore, the code can be defined as [8,4] Hamming code.

To decode the [8,4] Hamming code, first check the parity bit. If the parity bit indicates an error, single error correction (the [7,4] Hamming code) will indicate the error location, with "no error" indicating the parity bit. If the parity bit is correct, then single error correction will indicate the (bitwise) exclusive-or of two error locations. If the locations are equal ("no error") then a double bit error either has not occurred, or has cancelled itself out. Otherwise, a double bit error has occurred.