Hamming code

Hamming code

Hamming code is a set of error-correction code s that can be used to detect and correct bit errors that can occur when computer data is moved or stored. The parity-check matrix of a Hamming code is constructed by listing all columns of length that are non-zero, which means that the [[duacode of the Hamming code is the shortened Hadamard code. The parity-check matrix has the property that any two columns are pairwise linearly independent.

Due to the limited redundancy that Hamming codes add to the data, they can only detect and correct errors when the error rate is low. This is the case in computer memory (ECC memory), where bit errors are extremely rare and Hamming codes are widely used. In this context, an extended Hamming code having one extra parity bit is often used. Extended Hamming codes achieve a Hamming distance of four, which allows the decoder to distinguish between when at most one one-bit error occurs and when any two-bit errors occur. In this sense, extended Hamming codes are single-error correcting and double-error detecting, abbreviated as SECDED.

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Hamming and Hamming code theory

Richard Hamming, the inventor of Hamming codes, 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 punched paper tape, seven-eighths of an inch wide which had up to six holes per row. During weekdays, when errors in the relays were detected, the machine would stop and flash lights so that 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 detected errors. In a taped interview Hamming said, “And so I said, ‘Damn it, if the machine can detect an error, why can’t it locate the position of the error and correct it?'”. 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 today in applications such as ECC memory.

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 ones (bit-positions with values of one) in the preceding data was even or odd. If an odd number of bits is changed in transmission, the message will change parity and the error can be detected at this point; however, the bit that changed may have been the parity bit itself. The most common convention is that a parity value of one indicates that there is an odd number of ones in the data, and a parity value of zero indicates that there is an even number of ones. 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.

Two-out-of-five code

A two-out-of-five code is an encoding scheme which uses five bits consisting of exactly three 0s and two 1s. This provides ten possible combinations, enough to represent the digits 0–9. This scheme can detect all single bit-errors, all odd numbered bit-errors and some even numbered bit-errors (for example the flipping of both 1-bits). However it still cannot correct any of these errors.

Repetition

Another code in use at the time repeated every data bit multiple times in order to ensure that it was sent correctly. For instance, if the data bit to be sent is a 1, an repetition code will send 111. If the three bits received are not identical, an error occurred during transmission. 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, with the greater quantity of digits that are the same (‘0’ or a ‘1’) indicating what the data bit should be. A code with this ability to reconstruct the original message in the presence of errors is known as an error-correcting code. This triple repetition code is a Hamming code with since there are two parity bits, and data bit.

Such codes cannot correctly repair all errors, however. In our example, if the channel flips two bits and the receiver gets 001, the system will detect the error, but conclude that the original bit is 0, which is incorrect. If we increase the size of the bit string to four, we can detect all two-bit errors but cannot correct them, (the quantity of parity bits is even) at five bits, we can correct all two-bit errors, but not all three-bit errors.

Moreover, increasing the size of the parity bit string is 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 seven-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 seven bits, Hamming described this as an (8,7) code, with eight bits in total, of which seven are data. The repetition example would be (3,1), following the same logic. The code 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, so one bit flip can be detected, but not corrected and 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. It can correct one-bit errors or detect but not correct two-bit errors. A (4,1) repetition (each bit is repeated four times) has a distance of 4, so flipping three bits can be detected, but not corrected. When three bits flip in the same group there can be situations where attempting to correct will produce the wrong code word. In general, a code with distance k can detect but not correct errors.

Hamming was interested in two problems at once: increasing the distance as much as possible, while at the same time increasing the code 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 – Hamming Code Error Correction

The following general algorithm generates a single-error correcting (SEC) code for any number of bits.

  1. Number the bits starting from 1: bit 1, 2, 3, 4, 5, etc.
  2. Write the bit numbers in binary: 1, 10, 11, 100, 101, 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: bit 2 (the parity bit itself), 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.

The form of the parity is irrelevant. Even parity is mathematically simpler, but there is no difference in practice.

This general rule can be shown visually:

Hamming code

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.

As you can see, if you have parity bits, it can cover bits from 1 up to 2^m-1. If we subtract out the parity bits, we are left with 2^m-m-1 bits we can use for the data. As 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
…
n=2^m-1 k=2^m-m-1 Hamming(2^m-1,2^m-m-1) (2^m - m - 1)/(2^m-1)

If, in addition, an overall parity bit (bit 0) is included, the code can detect (but not correct) any two-bit error, making a SECDED code. The overall parity indicates whether the total number of errors is even or odd. If the basic Hamming code detects an error, but the overall parity says that there are an even number of errors, an uncorrectable 2-bit error has occurred.

Hamming codes with additional parity (SECDED)

Hamming codes have a minimum distance of 3, which means that the decoder can detect and correct a single error, but it cannot distinguish a double bit error of some codeword from a single bit error of a different codeword. Thus, some double-bit errors will be incorrectly decoded as if they were single bit errors and therefore go undetected, unless no correction is attempted.

To remedy this shortcoming, Hamming codes can be extended by an extra parity bit. This way, it is possible to increase the minimum distance of the Hamming code to 4, which allows the decoder to distinguish between single bit errors and two-bit errors. Thus the decoder can detect and correct a single error and at the same time detect (but not correct) a double error. If the decoder does not attempt to correct errors, it can detect up to three errors.

This extended Hamming code is popular in computer memory systems, where it is known as SECDED (abbreviated from 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 code 7-4

Graphical depiction of the four data bits and three parity bits and which parity bits apply to which data bits

In 1950, Hamming introduced the Hamming code 7-4. It encodes four data bits into seven bits by adding three parity bits. It can detect and correct single-bit errors. With the addition of an overall parity bit, it can also detect (but not correct) double-bit errors.

Construction of G and H

The matrix 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 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

mathbf{H},mathbf{G}^text{T} = mathbf{0}, an all-zeros matrix.

Since [7, 4, 3] = [n, k, d] = [2m − 1, 2m−1−m, m]. 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 mathbf{G} and the parity-check matrix mathbf{H} are:

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

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:.

The trick starts when a volunteer thinks of a number between 1 and 15 and a color, the magician shows colored cards with several numbers for the volunteer to say whether he sees his number, but he can lie on one card, according to the chosen color. The magician is able to find on which color the volunteer lied, and then tell the number.

Trick is based on the Hamming Code. The first four cards resemble a binary-digit trick with numbers 1 to 15. If there were no lie allowed, then we’d only need the first four cards. Simply, we would add the top left number on each card that faces up.

However, we are also trying to discover where the lie happened. This is similar to a computer system trying to fix a denigrated data communication. We need more digits (cards).

For every face-up card, you consider a 1 on the sequence. Face-down cards represent 0.

The first four cards will serve to compute the chosen number by adding the top-left number on cards facing up.

The last three cards are the “parity digits”.

Check the parity digits (last three cards, see below), if only one of them does not match, then that is the “lie”-card, and the chosen number is the sum of the digits on top left on the first four cards that face up;

If more than one parity digit does not match, then you follow the rule, according to discrepancy:

– Parity 1 and 2: the person lied on the card 3 (yellow), so you flip the card, and the chosen number will be the sum of the numbers on top left on the first four cards who face up (after you “fixed” the sequence);

– Parity 1 and 3: the person lied on the card 2 (orange), so you flip the card, and the chosen number will be the sum of the numbers on top left on the first four cards who face up (after you “fixed” the sequence);

– Parity 2 and 3: the person lied on the card 1 (red), so you flip the card, and the chosen number will be the sum of the numbers on top left on the first four cards who face up (after you “fixed” the sequence);

– Parity 1, 2 and 3: the person lied on the card 4 (green), so you flip the card, and the chosen number will be the sum of the numbers on top left on the first four cards who face up (after you “fixed” the sequence).

You are creating a sequence of seven digits 0’s and 1’s. Because there is a lie, the final sequence will not be any one on the table on “Error Detection” section. With multiple parity checks, we are able to identify the incorrect digit, fix it, and find the correspondent number.

This magic trick appears on an article by Ricardo Teixeira on the February 2017 edition of Math Horizon (“Decoding the Hamming Code,” Math Horizons, April 2006) and enhanced by Todd Mateer (“A Magic Trick Based on the Hamming Code,” Math Horizons, November 2013). Ehrenborg’s and Mateer’s tricks, while essentially the same as this one, require specially designed cards with a clever but little complicated system of tabs to aid the magician. The advantage and disadvantage of using simple rectangular cards is that the trick happens in your mind, instead of physically on the cards.

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Sources

http://wikipedia.org/