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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) .
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 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} .
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.
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 ...
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 bound is obtained by considering the range of parameters that are obtainable by concatenating a "good" outer code with a "good" inner code . Specifically, we suppose that the outer code meets the Singleton bound , i.e. it has rate r o u t {\displaystyle r_{out}} and relative distance δ o u t {\displaystyle \delta _{out}} satisfying r o u t ...
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