Hamming code


In computer science and telecommunications, Hamming codes are a family of linear error-correcting codes. Hamming codes can detect one-bit and two-bit errors, or correct one-bit errors without detection of uncorrected errors. By contrast, the simple parity code cannot correct errors, and can detect only an odd number of bits in error. Hamming codes are perfect codes, that is, they achieve the highest possible rate for codes with their block length and minimum distance of three.
Richard W. Hamming invented Hamming codes in 1950 as a way of automatically correcting errors introduced by punched card readers. In his original paper, Hamming elaborated his general idea, but specifically focused on the Hamming code which adds three parity bits to four bits of data.
In mathematical terms, Hamming codes are a class of binary linear code. For each integer there is a code-word with block length and message length. Hence the rate of Hamming codes is, which is the highest possible for codes with minimum distance of three and block length. The parity-check matrix of a Hamming code is constructed by listing all columns of length that are non-zero, which means that the dual code of the Hamming code is the shortened Hadamard code, also known as a Simplex 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, where bit errors are extremely rare and Hamming codes are widely used. Memory with this correction system is known as ECC memory. 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.

History

, the inventor of Hamming codes, worked at Bell Labs in the late 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 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 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. 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 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 ; at five bits, we can both detect and 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.

Description

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 ' code, with eight bits in total, of which seven are data. The repetition example would be ', 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". 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 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 it can detect - but not correct - two-bit errors. A repetition 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

The following general algorithm generates a single-error correcting code for any number of bits. The main idea is to choose the error-correcting bits such that the index-XOR is 0. We use positions 1, 10, 100, etc. 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 are parity bits: 1, 2, 4, 8, etc.
  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.
  6. # Parity bit 1 covers all bit positions which have the least significant bit set: bit 1, 3, 5, 7, 9, etc.
  7. # Parity bit 2 covers all bit positions which have the second least significant bit set: bits 2–3, 6–7, 10–11, etc.
  8. # Parity bit 4 covers all bit positions which have the third least significant bit set: bits 4–7, 12–15, 20–23, etc.
  9. # Parity bit 8 covers all bit positions which have the fourth least significant bit set: bits 8–15, 24–31, 40–47, etc.
  10. # 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 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:
Shown are only 20 encoded bits 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 parity bits, bits from 1 up to can be covered. After discounting the parity bits, bits remain for use as data. As varies, we get all the possible Hamming codes: