guava - Gap

GUAVA 34
IsDoublyEvenCode(C) returns ‘true’ if C is a binary linear code which has codewords of weight
divisible by 4 only. According to [HP03], a doubly-even code is self-orthogonal and every row in its
generator matrix has weight that is divisible by 4.
Example
gap> C:=BinaryGolayCode();
a cyclic [23,12,7]3 binary Golay code over GF(2)
gap> WeightDistribution(C);
[ 1, 0, 0, 0, 0, 0, 0, 253, 506, 0, 0, 1288, 1288, 0, 0, 506, 253, 0, 0, 0, 0, 0, 0, 1 ]
gap> IsDoublyEvenCode(C);
false
gap> C:=ExtendedCode(C);
a linear [24,12,8]4 extended code
gap> WeightDistribution(C);
[ 1, 0, 0, 0, 0, 0, 0, 0, 759, 0, 0, 0, 2576, 0, 0, 0, 759, 0, 0, 0, 0, 0, 0, 0, 1 ]
gap> IsDoublyEvenCode(C);
true
4.3.11 IsSinglyEvenCode
♦ IsSinglyEvenCode(C)
(function)
IsSinglyEvenCode(C) returns ‘true’ if C is a binary self-orthogonal linear code which is not
doubly-even. In other words, C is a binary self-orthogonal code which has codewords of even weight.
Example
gap> x:=Indeterminate(GF(2));
x_1
gap> C:=QuasiCyclicCode( [xˆ0, 1+xˆ3+xˆ5+xˆ6+xˆ7], 11, GF(2) );
a linear [22,11,1..6]4..7 quasi-cyclic code over GF(2)
gap> IsSelfDualCode(C); # self-dual is a restriction of self-orthogonal
true
gap> IsDoublyEvenCode(C);
false
gap> IsSinglyEvenCode(C);
true
4.3.12 IsEvenCode
♦ IsEvenCode(C)
(function)
IsEvenCode(C) returns ‘true’ if C is a binary linear code which has codewords of even weight–
regardless whether or not it is self-orthogonal.
Example
gap> C:=BinaryGolayCode();
a cyclic [23,12,7]3 binary Golay code over GF(2)
gap> IsSelfOrthogonalCode(C);
false
gap> IsEvenCode(C);
false