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For example, 1011 is encoded (using the non-systematic form of G at the start of this section) into 01 1 0 011 0 where blue digits are data; red digits are parity bits from the [7,4] Hamming code; and the green digit is the parity bit added by the [8,4] code. The green digit makes the parity of the [7,4] codewords even.
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 ...
Examples of block codes are Reed–Solomon codes, Hamming codes, Hadamard codes, Expander codes, Golay codes, Reed–Muller codes and Polar codes. These examples also belong to the class of linear codes, and hence they are called linear block codes. More particularly, these codes are known as algebraic block codes, or cyclic block codes ...
The Hamming(7,4) code is closely related to the E 7 lattice and, in fact, can be used to construct it, or more precisely, its dual lattice E 7 ∗ (a similar construction for E 7 uses the dual code [7,3,4] 2).
Examples of applications of the Hamming weight include: In modular exponentiation by squaring , the number of modular multiplications required for an exponent e is log 2 e + weight( e ). This is the reason that the public key value e used in RSA is typically chosen to be a number of low Hamming weight.
Here is a table of all n-bit lexicode by d-bit minimal hamming distance, resulting of maximum 2 m codewords dictionnary. For example, F 4 code (n=4,d=2,m=3), extended Hamming code (n=8,d=4,m=4) and especially Golay code (n=24,d=8,m=12) shows exceptional compactness compared to neighbors.
Gray code; Hamming codes. Hamming(7,4): a Hamming code that encodes 4 bits of data into 7 bits by adding 3 parity bits; Hamming distance: sum number of positions which are different; Hamming weight (population count): find the number of 1 bits in a binary word; Redundancy checks. Adler-32; Cyclic redundancy check; Damm algorithm; Fletcher's ...
English: Example Hamming(7,4) code of the data 1100 into 0111100 and extra parity bit 0. The parity of the red, green, blue, and yellow circles are all even (red, green, & blue have 2 1's; and yellow has 4 1's).