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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.
A perfect code may be interpreted as one in which the balls of Hamming radius t centered on codewords exactly fill out the space (t is the covering radius = packing radius). A quasi-perfect code is one in which the balls of Hamming radius t centered on codewords are disjoint and the balls of radius t +1 cover the space, possibly with some ...
The Gilbert–Varshamov bound for linear codes is related to the general Gilbert–Varshamov bound, which gives a lower bound on the maximal number of elements in an error-correcting code of a given block length and minimum Hamming weight over a field. This may be translated into a statement about the maximum rate of a code with given length ...
For a fixed length n, the Hamming distance is a metric on the set of the words of length n (also known as a Hamming space), as it fulfills the conditions of non-negativity, symmetry, the Hamming distance of two words is 0 if and only if the two words are identical, and it satisfies the triangle inequality as well: [2] Indeed, if we fix three words a, b and c, then whenever there is a ...
In the extended binary Golay code, all code words have Hamming weights of 0, 8, 12, 16, or 24. Code words of weight 8 are called octads and code words of weight 12 are called dodecads. Octads of the code G 24 are elements of the S(5,8,24) Steiner system. There are 759 = 3 × 11 × 23 octads and 759 complements thereof.
The Hamming scheme, named after Richard Hamming, is also known as the hyper-cubic association scheme, and it is the most important example for coding theory. [1] [2] [3] In this scheme =, the set of binary vectors of length , and two vectors , are -th associates if they are Hamming distance apart.
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 ...