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In mathematical terms, Hamming codes are a class of binary linear code. For each integer r ≥ 2 there is a code-word with block length n = 2r − 1 and message length k = 2r − r − 1. Hence the rate of Hamming codes is R = k / n = 1 − r / (2r − 1), which is the highest possible for codes with minimum distance of three (i.e., the minimal ...
Hamming (7,4) In coding theory, Hamming (7,4) is a linear error-correcting code that encodes four bits of data into seven bits by adding three parity bits. It is a member of a larger family of Hamming codes, but the term Hamming code often refers to this specific code that Richard W. Hamming introduced in 1950.
The Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string of bits, this is the number of 1's in the string, or the digit sum of the binary representation of a ...
Traditional Reed–Muller codes are binary codes, which means that messages and codewords are binary strings. When r and m are integers with 0 ≤ r ≤ m, the Reed–Muller code with parameters r and m is denoted as RM ( r , m ). When asked to encode a message consisting of k bits, where holds, the RM ( r , m) code produces a codeword ...
A block code (specifically a Hamming code) where redundant bits are added as a block to the end of the initial message A continuous convolutional code where redundant bits are added continuously into the structure of the code word. The two main categories of ECC codes are block codes and convolutional codes.
Convolutional codes are processed on a bit-by-bit basis. They are particularly suitable for implementation in hardware, and the Viterbi decoder allows optimal decoding. Block codes are processed on a block-by-block basis. Early examples of block codes are repetition codes, Hamming codes and multidimensional parity-check codes.
Such limitations often take the form of bounds that relate different parameters of the block code to each other, such as its rate and its ability to detect and correct errors. Examples of block codes are Reed–Solomon codes, Hamming codes, Hadamard codes, Expander codes, Golay codes, Reed–Muller codes and Polar codes.
It is a CSS code (Calderbank-Shor-Steane), using the classical binary [7,4,3] Hamming code to correct for both qubit flip errors (X errors) and phase flip errors (Z errors). The Steane code encodes one logical qubit in 7 physical qubits and is able to correct arbitrary single qubit errors. Its check matrix in standard form is