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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 ...
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
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 BCH code with and higher has the generator polynomial. This code has minimal Hamming distance 15 and corrects 7 errors. It has 1 data bit and 14 checksum bits. It is also denoted as: (15, 1) BCH code. In fact, this code has only two codewords: 000000000000000 and 111111111111111 (a trivial repetition code ).
The first element of a CIRC decoder is a relatively weak inner (32,28) Reed–Solomon code, shortened from a (255,251) code with 8-bit symbols. This code can correct up to 2 byte errors per 32-byte block. More importantly, it flags as erasures any uncorrectable blocks, i.e., blocks with more than 2 byte errors.
Turbo coding is an iterated soft-decoding scheme that combines two or more relatively simple convolutional codes and an interleaver to produce a block code that can perform to within a fraction of a decibel of the Shannon limit. Predating LDPC codes in terms of practical application, they now provide similar performance.
Hamming bound. In mathematics and computer science, in the field of coding theory, the Hamming bound is a limit on the parameters of an arbitrary block code: it is also known as the sphere-packing bound or the volume bound from an interpretation in terms of packing balls in the Hamming metric into the space of all possible words.
The only nontrivial and useful perfect codes are the distance-3 Hamming codes with parameters satisfying (2 r – 1, 2 r – 1 – r, 3), and the [23,12,7] binary and [11,6,5] ternary Golay codes. [4] [5] Another code property is the number of neighbors that a single codeword may have. [6] Again, consider pennies as an example.