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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 [ 1 ] proved by Joshi (1958) and even earlier by Komamiya (1953) .
The Singleton bound is that the sum of the rate and the relative distance of a block code cannot be much larger than 1: R + δ ≤ 1 + 1 n {\displaystyle R+\delta \leq 1+{\frac {1}{n}}} . In other words, every block code satisfies the inequality k + d ≤ n + 1 {\displaystyle k+d\leq n+1} .
Singleton pattern, a design pattern that allows only one instance of a class to exist; Singleton bound, used in coding theory; Singleton variable, a variable that is referenced only once; Singleton, a character encoded with one unit in variable-width encoding schemes for computer character sets
The Singleton bound states that the minimum distance d of a linear block code of size (n,k) is upper-bounded by n − k + 1. The distance d was usually understood to limit the error-correction capability to ⌊(d−1) / 2⌋. The Reed–Solomon code achieves this bound with equality, and can thus correct up to ⌊(n−k) / 2⌋ errors. However ...
Lemma (Singleton bound): Every linear [n,k,d] code C satisfies + +. A code C whose parameters satisfy k +d = n + 1 is called maximum distance separable or MDS. Such codes, when they exist, are in some sense best possible.
Folded Reed–Solomon codes and the singleton bound [ edit ] According to the asymptotic version of the singleton bound , it is known that the relative distance δ {\displaystyle \delta } , of a code must satisfy R ⩽ 1 − δ + o ( 1 ) {\displaystyle R\leqslant 1-\delta +o(1)} where R {\displaystyle R} is the rate of the code.
The first two of its ten chapters present background and introductory material, including Hamming distance, decoding methods including maximum likelihood and syndromes, sphere packing and the Hamming bound, the Singleton bound, and the Gilbert–Varshamov bound, and the Hamming(7,4) code.
an upper bound on the largest possible number of vertices in any graph with this degree and diameter. Therefore, these graphs solve the degree diameter problem for their parameters. Another equivalent definition of a Moore graph G is that it has girth g = 2 k + 1 and precisely n / g ( m – n + 1) cycles of length g , where n and m are ...