In
coding theory
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and computer data storage, data sto ...
, the Singleton bound, named after the American mathematician Richard Collom Singleton (1928–2007), is a relatively crude upper bound on the size of an arbitrary
block code
In coding theory, block codes are a large and important family of Channel coding, error-correcting codes that encode data in blocks.
There is a vast number of examples for block codes, many of which have a wide range of practical applications. Th ...
with block length
, size
and minimum distance
. It is also known as the Joshibound proved by and even earlier by
Komamiya.
Statement of the bound
The minimum distance of a set
of codewords of length
is defined as
where
is the
Hamming distance
In information theory, the Hamming distance between two String (computer science), strings or vectors of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number ...
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
Proof
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
Puncture, punctured or puncturing may refer to:
* a flat tyre in British English (US English "flat tire" or just "flat")
* a penetrating wound caused by pointy objects as nails or needles
* Lumbar puncture, also known as a spinal tap
* 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
In information theory, the Hamming distance between two String (computer science), strings or vectors of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number ...
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:
Linear codes
If
is a
linear code
In coding theory, a linear code is an error-correcting code for which any linear combination of Code word (communication), codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although t ...
with block length
, dimension
and minimum distance
over the
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
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
In coding theory, a parity-check matrix of a linear block code ''C'' is a matrix which describes the linear relations that the components of a codeword must satisfy. It can be used to decide whether a particular vector is a codeword and is also u ...
is
. Another simple proof follows from observing that the rows of any generator matrix in standard form have weight at most
.
History
The usual citation given for this result is , but it was proven earlier by . Joshi notes that the result had been obtained earlier by using a more complex proof. also notes the same regarding .
MDS codes
Linear block codes that achieve equality in the Singleton bound are called MDS (maximum distance separable) codes. Examples of such codes include codes that have only
codewords (the all-
word for
, having thus minimum distance
), codes that use the whole of
(minimum distance 1), codes with a single parity symbol (minimum distance 2) and their
dual code
In coding theory, the dual code of a linear code
:C\subset\mathbb_q^n
is the linear code defined by
:C^\perp = \
where
:\langle x, c \rangle = \sum_^n x_i c_i
is a scalar product. In linear algebra terms, the dual code is the annihilator ...
s. These are often called ''trivial'' MDS codes.
In the case of binary alphabets, only trivial MDS codes exist.
Examples of non-trivial MDS codes include
Reed-Solomon codes and their extended versions.
MDS codes are an important class of block codes since, for a fixed
and
, they have the greatest error correcting and detecting capabilities. There are several ways to characterize MDS codes:
The last of these characterizations permits, by using the
MacWilliams identities, an explicit formula for the complete weight distribution of an MDS code.
Arcs in projective geometry
The linear independence of the columns of a generator matrix of an MDS code permits a construction of MDS codes from objects in
finite
Finite may refer to:
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect
* "Finite", a song by Sara Gr ...
projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...
. Let
be the finite
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
of (geometric) dimension
over the finite field
. Let
be a set of points in this projective space represented with
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...
. Form the
matrix
whose columns are the homogeneous coordinates of these points. Then,
See also
*
Gilbert–Varshamov bound In coding theory, the Gilbert–Varshamov bound (due to Edgar Gilbert and independently Rom Varshamov.) is a bound on the size of a (not necessarily linear) code. It is occasionally known as the Gilbert– Shannon–Varshamov bound (or the GSV bou ...
*
Griesmer bound
*
Hamming bound
In 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 Spher ...
*
Johnson bound
*
Plotkin bound
Notes
References
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Further reading
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{{DEFAULTSORT:Singleton Bound
Coding theory
Inequalities (mathematics)
Articles containing proofs