Singleton bound

In coding theory, the Singleton bound, named after Richard Collom Singleton, is a relatively crude upper bound on the size of an arbitrary block code with block length , size and minimum distance . It is also known as the Joshibound. proved by and even earlier by . The minimum distance of a set of codewords of length is defined as where is the Hamming distance between and . The expression represents the maximum number of possible codewords in a -ary block code of length and minimum distance . Then the Singleton bound states that First observe that the number of -ary words of length is , since each letter in such a word may take one of different values, independently of the remaining letters. Now let be an arbitrary -ary block code of minimum distance . Clearly, all codewords are distinct. If we puncture the code by deleting the first letters of each codeword, then all resulting codewords must still be pairwise different, since all of the original codewords in have Hamming distance at least from each other. Thus the size of the altered code is the same as the original code. The newly obtained codewords each have length and thus, there can be at most of them. Since was arbitrary, this bound must hold for the largest possible code with these parameters, thus: If is a linear code with block length , dimension and minimum distance over the finite field with elements, then the maximum number of codewords is and the Singleton bound implies: so that which is usually written as In the linear code case a different proof of the Singleton bound can be obtained by observing that rank of the parity check matrix is . Another simple proof follows from observing that the rows of any generator matrix in standard form have weight at most . The usual citation given for this result is , but was proven earlier by . Joshi notes that the result was obtained earlier by using a more complex proof. also notes the same regarding . Linear block codes that achieve equality in the Singleton bound are called MDS (maximum distance separable) codes.

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Concepts associés (4)

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Code parfait et code MDS

Les codes parfaits et les codes à distance séparable maximale (MDS), sont des types de codes correcteurs d'erreur. Un code correcteur est un code permettant au récepteur de détecter ou de corriger des altérations à la suite de la transmission ou du stockage. Elle est rendue possible grâce à une redondance de l'information. Un code est dit parfait s'il ne contient aucune redondance inutile. Le concept correspond à un critère d'optimalité. Un code est dit MDS s'il vérifie un autre critère d'optimalité s'exprimant dans le contexte des codes linéaires.

Griesmer bound

In 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 bound

In 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 .

Hedging pour les LPs CS-450: Algorithms II

Couvre l'algorithme Hedge pour minimiser les pertes dans les problèmes de programmation linéaire.