Definitions

# Hamming code

In telecommunication, a Hamming code is a linear error-correcting code named after its inventor, Richard Hamming. Hamming codes can detect and correct single-bit errors. In other words, the Hamming distance between the transmitted and received code-words must be zero or one for reliable communication. Alternatively, it can detect (but not correct) up to two simultaneous bit errors.

In contrast, the simple parity code cannot correct errors, nor can it be used to detect more than one error (such as where two bits are transposed).

In mathematical terms, Hamming codes are a class of binary linear codes. For each integer $m>1$ there is a code with parameters: $\left[2^m-1, 2^m-m-1, 3\right]$. The parity-check matrix of a Hamming code is constructed by listing all columns of length $m$ that are pair-wise independent.

Because of the simplicity of Hamming codes, they are widely used in computer memory (RAM). In particular, a single-error-correcting and double-error-detecting variant commonly referred to as SECDED.

## History

Hamming worked at Bell Labs in the 1940s on the Bell Model V computer, an electromechanical relay-based machine with cycle times in seconds. Input was fed in on punch cards, which would invariably have read errors. During weekdays, special code would find errors and flash lights so the operators could correct the problem. During after-hours periods and on weekends, when there were no operators, the machine simply moved on to the next job.

Hamming worked on weekends, and grew increasingly frustrated with having to restart his programs from scratch due to the unreliability of the card reader. Over the next few years he worked on the problem of error-correction, developing an increasingly powerful array of algorithms. In 1950 he published what is now known as Hamming Code, which remains in use in some applications today.

### 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.

#### Parity

Parity adds a single bit that indicates whether the number of 1 bits in the preceding data was even or odd. If a single bit is changed in transmission, the message will change parity and the error can be detected at this point. (Note that the bit that changed may have been the parity bit itself!) The most common convention is that a parity value of 1 indicates that there is an odd number of ones in the data, and a parity value of 0 indicates that there is an even number of ones in the data. In other words: The data and the parity bit together should contain an even number of 1s.

Parity checking is not very robust, since if the number of bits changed is even, the check bit will be valid and the error will not be detected. Moreover, parity does not indicate which bit contained the error, even when it can detect it. The data must be discarded entirely and re-transmitted from scratch. On a noisy transmission medium, a successful transmission could take a long time or may never occur. However, while the quality of parity checking is poor, since it uses only a single bit, this method results in the least overhead. Furthermore, parity checking does allow for the restoration of a missing bit when the missing bit is known.

#### Two-out-of-five code

In the 1940s Bell used a slightly more sophisticated m of n code known as the two-out-of-five code. This code ensured that every block of five bits (known as a 5-block) had exactly two 1s. The computer could tell there was an error if, in its input, there were not exactly two 1s in each block. Two-of-five was still only able to detect single bit errors; if one bit flipped to a 1 and another to a 0 in the same block, the two-of-five rule remained true and the error would go undiscovered.

#### Repetition

Another code in use at the time repeated every data bit several times in order to ensure that it got through. For instance, if the data bit to be sent was a 1, an n=3 repetition code would send "111". If the three bits received were not identical, an error occurred. If the channel is clean enough, most of the time only one bit will change in each triple. Therefore, 001, 010, and 100 each correspond to a 0 bit, while 110, 101, and 011 correspond to a 1 bit, as though the bits counted as "votes" towards what the original bit was. A code with this ability to reconstruct the original message in the presence of errors is known as an error-correcting code.

Such codes cannot correctly repair all errors, however. In our example, if the channel flipped two bits and the receiver got "001", the system would detect the error, but conclude that the original bit was 0, which is incorrect. If we increase the number of times we duplicate each bit to four, we can detect all two-bit errors but can't correct them (the votes "tie"); at five, we can correct all two-bit errors, but not all three-bit errors.

Moreover, the repetition code is extremely inefficient, reducing throughput by three times in our original case, and the efficiency drops drastically as we increase the number of times each bit is duplicated in order to detect and correct more errors.

## Hamming codes

If more error-correcting bits are included with a message, and if those bits can be arranged such that different incorrect bits produce different error results, then bad bits could be identified. In a 7-bit message, there are seven possible single bit errors, so three error control bits could potentially specify not only that an error occurred but also which bit caused the error.

Hamming studied the existing coding schemes, including two-of-five, and generalized their concepts. To start with he developed a nomenclature to describe the system, including the number of data bits and error-correction bits in a block. For instance, parity includes a single bit for any data word, so assuming ASCII words with 7-bits, Hamming described this as an (8,7) code, with eight bits in total, of which 7 are data. The repetition example would be (3,1), following the same logic. The information rate is the second number divided by the first, for our repetition example, 1/3.

Hamming also noticed the problems with flipping two or more bits, and described this as the "distance" (it is now called the Hamming distance, after him). Parity has a distance of 2, as any two bit flips will be invisible. The (3,1) repetition has a distance of 3, as three bits need to be flipped in the same triple to obtain another code word with no visible errors. A (4,1) repetition (each bit is repeated four times) has a distance of 4, so flipping two bits can be detected, but not corrected. When three bits flip in the same group there can be situations where the code corrects towards the wrong code word.

Hamming was interested in two problems at once; increasing the distance as much as possible, while at the same time increasing the information rate as much as possible. During the 1940s he developed several encoding schemes that were dramatic improvements on existing codes. The key to all of his systems was to have the parity bits overlap, such that they managed to check each other as well as the data.

### General algorithm

Although any number of algorithms can be created, the following general algorithm positions the parity bits at powers of two to ease calculation of which bit was flipped upon detection of incorrect parity.

1. All bit positions that are powers of two are used as parity bits. (positions 1, 2, 4, 8, 16, 32, 64, etc.), see at the On-Line Encyclopedia of Integer Sequences.
2. All other bit positions are for the data to be encoded. (positions 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, etc.), see at the On-Line Encyclopedia of Integer Sequences.
3. Each parity bit calculates the parity for some of the bits in the code word. The position of the parity bit determines the sequence of bits that it alternately checks and skips.
• Position 1 (n=1): skip 0 bit (0=n−1), check 1 bit (n), skip 1 bit (n), check 1 bit (n), skip 1 bit (n), etc. (1,3,5,7,9,11,13,15,...)
• Position 2 (n=2): skip 1 bit (1=n−1), check 2 bits (n), skip 2 bits (n), check 2 bits (n), skip 2 bits (n), etc. (2,3,6,7,10,11,14,15,...)
• Position 4 (n=4): skip 3 bits (3=n−1), check 4 bits (n), skip 4 bits (n), check 4 bits (n), skip 4 bits (n), etc. (4,5,6,7,12,13,14,15,20,21,22,23,...)
• Position 8 (n=8): skip 7 bits (7=n−1), check 8 bits (n), skip 8 bits (n), check 8 bits (n), skip 8 bits (n), etc. (8-15,24-31,40-47,...)
• Position 16 (n=16): skip 15 bits (15=n−1), check 16 bits (n), skip 16 bits (n), check 16 bits (n), skip 16 bits (n), etc. (16-31,48-63,80-95,...)
• Position 32 (n=32): skip 31 bits (31=n−1), check 32 bits (n), skip 32 bits (n), check 32 bits (n), skip 32 bits (n), etc. (32-63,96-127,160-191,...)
• General rule for position n: skip n−1 bits, check n bits, skip n bits, check n bits...
• And so on.

In other words, the parity bit at position $2^k$ checks bits in positions having bit k set in their binary representation. Conversely, for instance, bit 13, i.e. 1101(2), is checked by bits 1000(2) = 8, 0100(2)=4 and 0001(2) = 1.

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 p3 d2 d3 d4 p4 d5 d6 d7 d8 d9 d10 d11 p5 d12 d13 d14 d15
Parity
bit
coverage
p1 X X X X X X X X X X
p2 X X X X X X X X X X
p3 X X X X X X X X X
p4 X X X X X X X X
p5 X X X X X

Shown are only 20 encoded bits (5 parity, 15 data) but the pattern continues indefinitely. The key thing about Hamming Codes that can easily be seen from visual inspection is that any given bit has a unique parity bit coverage. For example, the only bit covered by p3 and p4 only is bit 12 (d8). It is this unique bit coverage that lets a Hamming Code correct any single-bit error, and at the same time allows the code to detect (but not correct) any two-bit error. For example, if bits 1 (p1) & 2 (p2) were flipped then this would be confused with bit 3 (d1) being flipped since the parity bit coverage of bit 3 is the same as bits 1 & 2.

## Hamming codes with additional parity (SECDED)

Hamming codes have a minimum distance of 3, which means that the code can detect and correct a single error, but a double bit error is indistinguishable from a different code with a single bit error. Thus, they can detect double-bit errors only if correction is not attempted.

By including an extra parity bit, it is possible to increase the minimum distance of the Hamming code to 4. This gives the code the ability to detect and correct a single error and at the same time detect (but not correct) a double error. (It could also be used to detect up to 3 errors but not correct any.)

This code system is popular in computer memory systems, where it is known as SECDED ("single error correction, double error detection"). Particularly popular is the (72,64) code, a truncated (127,120) Hamming code plus an additional parity bit, which has the same space overhead as a (9,8) parity code.

## Hamming(7,4) code

In 1950, Hamming introduced the (7,4) code. It encodes 4 data bits into 7 bits by adding three parity bits. Hamming(7,4) can detect and correct single-bit errors but can only detect double-bit errors.

## Construction of G and H

The matrix $mathbf\left\{G\right\} := begin\left\{pmatrix\right\} I_k | -A^T end\left\{pmatrix\right\}$ is called a (Canonical) generator matrix of a linear (n,k) code,

and $mathbf\left\{H\right\} := begin\left\{pmatrix\right\} A | I_\left\{n-k\right\} end\left\{pmatrix\right\}$ 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

$mathbf\left\{H\right\},mathbf\left\{G\right\}^T = mathbf\left\{0\right\}$, an all-zeros matrix [Moon, p.89].

Since (7,4,3)=(n,k,d)=[2m − 1, 2m−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 right side is all of the nonzero n-tuples where order of the n-tuples in the columns of matrix does not matter. The left 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 right hand side.

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

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

and

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

Finally, these matrices can be mutated into equivalent non-systematic codes by the following operations [Moon, p. 85]:

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

## Encoding

Example

From the above matrix we have 2k=24=16 codewords. The codewords $overrightarrow\left\{x\right\}$ of this binary code can be obtained from $overrightarrow\left\{x\right\}=overrightarrow\left\{a\right\}G$. With $overrightarrow\left\{a\right\}=a_1a_2a_3a_4$ with $a_i$ exist in $F_2$ (A field with two elements namely 0 and 1).

Thus the codewords are all the 4-tuples (k-tuples).

Therefore,

(1,0,1,1) gets encoded as (1,0,1,1,0,1,0).

## Hamming(8,4) code

The Hamming(7,4) 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:

$mathbf\left\{G\right\} := begin\left\{pmatrix\right\}$
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}_{8,4}

and


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:


mathbf{H} = left(left.begin{array}{cccc} 0 & 1 & 1 & 1 1 & 0 & 1 & 1 1 & 1 & 0 & 1 1 & 1 & 1 & 0end{array}right|begin{array}{cccc} 1 & 0 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 0 & 0 & 0 & 1end{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


mathbf{G} = left(left.begin{array}{cccc} 1 & 0 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 0 & 0 & 0 & 1end{array}right|begin{array}{cccc} 0 & 1 & 1 & 1 1 & 0 & 1 & 1 1 & 1 & 0 & 1 1 & 1 & 1 & 0end{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 into 01100110 where blue digits are data; red digits are parity from the Hamming(7,4) code; and the green digit is the parity added by Hamming(8,4). The green digit makes the parity of the (7,4) code even.

Finally, it can be shown that the minimum distance has increased from 3, as with the (7,4) code, to 4 with the (8,4) code. Therefor, the code can be defined as Hamming(8,4,4).

## Hamming(11,7) code

Consider the 7-bit data word "0110101". To demonstrate how Hamming codes are calculated and used to detect an error, see the tables below. They use d to signify data bits and p to signify parity bits.

Firstly the data bits are inserted into their appropriate positions and the parity bits calculated in each case using even parity. The diagram to the right shows which of the four parity bits cover which data bits.

Calculation of Hamming code parity bits
p1 p2 d1 p3 d2 d3 d4 p4 d5 d6 d7
Data word (without parity): 0 1 1 0 1 0 1
p1 1 0 1 0 1 1
p2 0 0 1 0 0 1
p3 0 1 1 0
p4 0 1 0 1
Data word (with parity): 1 0 0 0 1 1 0 0 1 0 1

The new data word (with parity bits) is now "10001100101". We now assume the final bit gets corrupted and turned from 1 to 0. Our new data word is "10001100100"; and this time when we analyze how the Hamming codes were created we flag each parity bit as 1 when the even parity check fails.

Checking of parity bits (switched bit highlighted)
p1 p2 d1 p3 d2 d3 d4 p4 d5 d6 d7 Parity check Parity bit
Received data word: 1 0 0 0 1 1 0 0 1 0 0
p1 1 0 1 0 1 0 Fail 1
p2 0 0 1 0 0 0 Fail 1
p3 0 1 1 0 Pass 0
p4 0 1 0 0 Fail 1

The final step is to evaluate the value of the parity bits (remembering the bit with lowest index is the least significant bit, i.e., it goes furthest to the right). The integer value of the parity bits is 11, signifying that the 11th bit in the data word (including parity bits) is wrong and needs to be flipped.

p4 p3 p2 p1
Binary 1 0 1 1
Decimal 8 2 1 Σ = 11

Flipping the 11th bit changes 10001100100 back into 10001100101. Removing the Hamming codes gives the original data word of 0110101.

Note that as parity bits do not check each other, if a single parity bit check fails and all others succeed, then it is the parity bit in question that is wrong and not any bit it checks.

Finally, suppose two bits change, at positions x and y. If x and y have the same bit at the 2k position in their binary representations, then the parity bit corresponding to that position checks them both, and so will remain the same. However, some parity bit must be altered, because xy, and so some two corresponding bits differ in x and y. Thus, the Hamming code detects all two bit errors — however, it cannot distinguish them from 1-bit errors.

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