MinimumDistance( C )
MinimumDistance
returns the minimum distance of C, the largest
integer d with the property that every element of C has at least a
Hamming distance d (see DistanceCodeword) to any other element of
C. For linear codes, the minimum distance is equal to the minimum
weight. This means that d is also the smallest positive value with
w[d+1] neq 0, where w is the weight distribution of C (see
WeightDistribution). For unrestricted codes, d is the smallest
positive value with w[d+1] neq 0, where w is the inner distribution
of C (see InnerDistribution).
For codes with only one element, the minimum distance is defined to be equal to the word length.
gap> C := MOLSCode(7);; MinimumDistance(C); 3 gap> WeightDistribution(C); [ 1, 0, 0, 24, 24 ] gap> MinimumDistance( WholeSpaceCode( 5, GF(3) ) ); 1 gap> MinimumDistance( NullCode( 4, GF(2) ) ); 4 gap> C := ConferenceCode(9);; MinimumDistance(C); 4 gap> InnerDistribution(C); [ 1, 0, 0, 0, 63/5, 9/5, 18/5, 0, 9/10, 1/10 ]
MinimumDistance( C, w )
In this form, MinimumDistance
returns the minimum distance of a
codeword w to the code C, also called the distance to C. This is
the smallest value d for which there is an element c of the code C
which is at distance d from w. So d is also the minimum value for
which D[d+1] neq 0, where D is the distance distribution of w to
C (see DistancesDistribution).
Note that w must be an element of the same vector space as the elements of C. w does not necessarily belong to the code (if it does, the minimum distance is zero).
gap> C := MOLSCode(7);; w := CodewordNr( C, 17 ); [ 2 2 4 6 ] gap> MinimumDistance( C, w ); 0 gap> C := RemovedElementsCode( C, w );; MinimumDistance( C, w ); 3 # so w no longer belongs to C
GAP 3.4.4