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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, also known as a Simplex code. The parity-check matrix has the property that any two columns are pairwise linearly independent.
then resemblance to rows 1, 2, and 4 of the code generator matrix (G) below will also be evident. So, by picking the parity bit coverage correctly, all errors with a Hamming distance of 1 can be detected and corrected, which is the point of using a Hamming code.
Codes in general are often denoted by the letter C, and a code of length n and of rank k (i.e., having n code words in its basis and k rows in its generating matrix) is generally referred to as an (n, k) code. Linear block codes are frequently denoted as [n, k, d] codes, where d refers to the code's minimum Hamming distance between any two code ...
For general , the generator matrix of the augmented Hadamard code is a parity-check matrix for the extended Hamming code of length and dimension , which makes the augmented Hadamard code the dual code of the extended Hamming code. Hence an alternative way to define the Hadamard code is in terms of its parity-check matrix: the parity-check ...
Type II codes are binary self-dual codes which are doubly even. Type III codes are ternary self-dual codes. Every codeword in a Type III code has Hamming weight divisible by 3. Type IV codes are self-dual codes over F 4. These are again even. Codes of types I, II, III, or IV exist only if the length n is a multiple of 2, 8, 4, or 2 respectively.
A code which attains the Hamming bound is said to be a perfect code. Hamming codes are perfect codes. [13] [14] Returning to differential equations, Hamming studied means of numerically integrating them. A popular approach at the time was Milne's Method, attributed to Arthur Milne. [15]
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
A generalisation of the technique used by Steane, to develop the 7-qubit code from the classical [7, 4] Hamming code, led to the construction of an important class of codes called the CSS codes, named for their inventors: Robert Calderbank, Peter Shor and Andrew Steane. According to the quantum Hamming bound, encoding a single logical qubit and ...