Hamming boundIn 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. It gives an important limitation on the efficiency with which any error-correcting code can utilize the space in which its code words are embedded. A code that attains the Hamming bound is said to be a perfect code.
Griesmer boundIn the mathematics of coding theory, the Griesmer bound, named after James Hugo Griesmer, is a bound on the length of linear binary codes of dimension k and minimum distance d. There is also a very similar version for non-binary codes. For a binary linear code, the Griesmer bound is: Let denote the minimum length of a binary code of dimension k and distance d. Let C be such a code. We want to show that Let G be a generator matrix of C. We can always suppose that the first row of G is of the form r = (1, ...
Plotkin boundIn the mathematics of coding theory, the Plotkin bound, named after Morris Plotkin, is a limit (or bound) on the maximum possible number of codewords in binary codes of given length n and given minimum distance d. A code is considered "binary" if the codewords use symbols from the binary alphabet . In particular, if all codewords have a fixed length n, then the binary code has length n. Equivalently, in this case the codewords can be considered elements of vector space over the finite field .
Johnson boundIn applied mathematics, the Johnson bound (named after Selmer Martin Johnson) is a limit on the size of error-correcting codes, as used in coding theory for data transmission or communications. Let be a q-ary code of length , i.e. a subset of . Let be the minimum distance of , i.e. where is the Hamming distance between and . Let be the set of all q-ary codes with length and minimum distance and let denote the set of codes in such that every element has exactly nonzero entries. Denote by the number of elements in .
