Hamming Code

Introduction

Hamming code is a specific type of error-detecting and correcting code invented by Richard Hamming in 1950. It is a linear block code that has been widely used in computer systems to detect and correct single-bit errors in stored and transmitted data. The Hamming code is a fundamental concept in the field of information theory and coding theory.

History

Richard Hamming, a mathematician and computer scientist, invented the Hamming code while working at Bell Labs. He was inspired to create an error-detection and correction code after experiencing the frequent errors that occurred in the mechanical computers of the era. Hamming's work on error-detection and correction codes marked a significant advancement in the field of information theory and has had a profound impact on the development of digital communication and storage systems.

Principle of Operation

Hamming code operates on the principle of parity check. It uses extra "parity bits" to represent the parity of a set of bits, which can then be used to detect and correct errors. The parity bits are placed in positions that are powers of 2, starting from the least significant bit (position 1). The remaining positions are filled with the data bits. 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 checks.

Mathematical Description

The Hamming code can be described mathematically as a (7,4) code, meaning it encodes 4 bits of data into 7 bits by adding three parity bits. It can detect up to two simultaneous bit errors, and correct single-bit errors, hence, it is considered a single-error correction, double-error detection (SECDED) code. The parity-check matrix of a Hamming code is constructed by listing all columns of length r that are non-zero, which means that the dual code of the Hamming code is the shortened Hadamard code.

Hamming Distance

The concept of Hamming distance is central to the functioning of the Hamming code. Hamming distance is a metric for comparing two binary data strings. While comparing, if the ith bit is not same in both strings, the Hamming distance is increased by one. The Hamming distance in a Hamming code is defined as the minimum number of bit changes required to go from any valid code word to any other valid code word. Hamming codes have a minimum Hamming distance of three, which allows for error detection and correction as previously described.

Hamming(7,4) Code

The Hamming(7,4) code is a linear block code that encodes 4 bits of data into 7 bits by adding three parity bits. It uses four data bits and three parity bits, which are inserted into the data bits at positions that are powers of two. The Hamming(7,4) code can detect up to two simultaneous bit errors, and correct single-bit errors.

Applications

Hamming codes have been widely used in computer systems and other digital systems to detect and correct single-bit errors. They are used in various forms of storage such as RAM, FLASH memory, and hard disk drives. They are also used in wireless communication to ensure data integrity. Hamming codes are even used in the implementation of error detection and correction in the network layer of the OSI model.

Limitations

While Hamming codes are effective for single-bit error detection and correction, they are not suitable for systems that experience burst errors or multiple bit errors. For such systems, more complex error detection and correction codes such as Reed-Solomon codes or convolutional codes are used.

References

Hamming, R. W. (1950). Error detecting and error correcting codes. Bell System technical journal, 29(2), 147-160.
Moon, Todd K. (2005). Error Correction Coding: Mathematical Methods and Algorithms. Wiley-Interscience. ISBN 0-471-64800-0.