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Magma supplies various functions for computing lower and upper bounds for
parameters associated with codes. It also contains tables of best known
bounds for linear codes.
The functions in this section only apply to codes over finite fields.
A Magma database allows the user access to tables giving
the best known upper and lower bounds of the Length,
Dimension, and MinimumWeight of linear codes.
Tables are currently available relating to codes
over GF(2) and GF(4) with 1 ≤ Length ≤256,
over GF(3) with 1 ≤ Length ≤243,
over GF(5), GF(8), and GF(9) with 1 ≤ Length ≤130,
and over GF(7) with 1 ≤ Length ≤100.
Returns the best known lower bound on the maximum possible
minimum weight of a linear code over finite field F having length
n and dimension k.
Returns the best known upper bound on the minimum weight of a linear
code over finite field F of length n and dimension k.
Returns the best known lower bound on the minimum possible length
of a linear code over finite field F having dimension k and
minimum weight at least d.
If the required length is out of the range of the database then
no bound is available and -1 is returned.
Returns the best known upper bound on the minimum possible length of
a linear code over finite field F of dimension k and minimum weight
at least d.
If the required length is out of the range of the database then
no bound is available and -1 is returned.
Returns the best known lower bound on the maximum possible
dimension of a linear code over finite field F having length n
and minimum weight at least d.
Returns the best known upper bound on the dimension of
a linear code over finite field F having length n
and minimum weight at least d.
Return the Elias upper bound of the cardinality of a largest code of
length n and minimum distance d over the field K.
Return the Griesmer upper bound of the cardinality of a largest code of
length n and minimum distance d over the field K.
Return the Johnson upper bound of the cardinality of a largest binary code of
length n and minimum distance d.
Return the Levenshtein upper bound of the cardinality of a largest code of
length n and minimum distance d over the field K.
Return the Plotkin upper bound on the cardinality of a (possibly non-linear)
code of length n and minimum distance d over the field K.
The bound is formed by calculating
the maximal possible average distance between codewords.
For binary codes the bound exists for n ≤2d, (d even), or
n ≤2d + 1 (d odd).
For codes over general fields the bound exists for
d > (1 - 1/#K) * n .
Return the Singleton upper bound of the cardinality of a largest code of
length n and minimum distance d over the field K.
Return the Hamming sphere packing upper bound on the cardinality of a
largest codes of length n and minimum distance d over the field K.
Return the Gilbert--Varshamov lower bound of the cardinality of a largest code
(possibly non-linear) of length n and minimum distance d over the field K.
Return the Gilbert--Varshamov lower bound of the cardinality of a largest linear
code of length n and minimum distance d over the field K.
Return the van Lint lower bound of the cardinality of a largest code of
length n and minimum distance d over the field K.
We compare computed and stored values of best known upper
bounds of the dimension
of binary linear codes of length 20. The cardinality of a linear code
of dimension k over GF(q) is q k, and so the computed bounds on
cardinality are compared with the stored bounds on dimension by taking
logs.
> n:=20;
> K := GF(2);
> [ Ilog(#K, Minimum({GriesmerBound(K, n, d), EliasBound(K, n, d),
> JohnsonBound(n, d) , LevenshteinBound(K, n, d),
> SpherePackingBound(K, n, d)})) : d in [1..n] ];
[ 20, 19, 15, 14, 12, 11, 9, 8, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1 ]
> [ BDLCUpperBound(K, n, d) : d in [1..n] ];
[ 20, 19, 15, 14, 11, 10, 9, 8, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1 ]
Given a cyclic code C, return the BCH bound for C. This a lower
bound on the minimum weight of C.
Return the Griesmer upper bound of the minimum weight of a linear code of
length n and dimension k over the field K.
Return the Elias asymptotic upper bound of the information rate for
δ in [0, 1] over the field K.
Return the McEliece--Rodemich--Rumsey--Welch asymptotic upper bound of the
binary information rate for δ in [0, 1].
Return the Plotkin asymptotic upper bound of the information rate for
δ in [0, 1] over the field K.
Return the Singleton asymptotic upper bound of the information rate for
δ in [0, 1] over any finite field.
Return the Hamming asymptotic upper bound of the information rate
for δ in [0, 1] over the field K.
Return the Gilbert--Varshamov asymptotic lower bound of the information rate
for δ in [0, 1] over the field K.
Return the Griesmer lower bound of the length of a linear code of
dimension k and minimum distance d over K.
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