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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.
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).
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 ...
Codes that attain the Hamming bound are called perfect codes. Examples include codes that have only one codeword, and codes that are the whole of A q n {\displaystyle \scriptstyle {\mathcal {A}}_{q}^{n}} .
A code is defined to be equidistant if and only if there exists some constant d such that the distance between any two of the code's distinct codewords is equal to d. [4] In 1984 Arrigo Bonisoli determined the structure of linear one-weight codes over finite fields and proved that every equidistant linear code is a sequence of dual Hamming ...
A typical example of linear code is the Hamming code. Codes defined via a Hamming space necessarily have the same length for every codeword, so they are called block codes when it is necessary to distinguish them from variable-length codes that are defined by unique factorization on a monoid.
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.