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.
> 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.