Some simple codes

Now we start to investigate codes in GAP. This requires the package GUAVA. To load GUAVA, type RequirePackage(``guava'');. GAP will then display the GUAVA banner:

   ____                          |
  /            \           /   --+--  Version 1.5
 /      |    | |\\        //|    |
|    _  |    | | \\      // |     Jasper Cramwinckel
|     \ |    | |--\\    //--|     Erik Roijackers
 \     ||    | |   \\  //   |     Reinald Baart
  \___/  \___/ |    \\//    |     Eric Minkes
                                  Lea Ruscio
true

How do you enter a code it using only the list of codes words? Easy, here's an example:

gap> C:=ElementsCode(["0000","1111"],"repetition code",GF(2));
a (4,2,1..4)2 repetition code over GF(2)

This notation (4,2,1..4)2 is GUAVA shorthand for: the length is 4, the size is 2, the minimum distance in the Hamming metric is between 1 and 4, and the covering radius in the Hamming metric is 2.

Here's how to check all this is correct:

gap> Elements(C);
[ [ 0 0 0 0 ], [ 1 1 1 1 ] ]
gap> MinimumDistance(C);
4
gap> Dimension(C);
1

GUAVA didn't know the minimum distance before, but once you type the MinimumDistance(C); command it modifies the GAP record for $C$.

gap> C;
a cyclic [4,1,4]2 repetition code over GF(2)

Let us not enter a code using a generator matrix $G$:

gap> G := Z(2)*[ [1,0,0,1,1,0], [0,1,0,1,1,0], [0,0,1,0,1,1] ];
[ [ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) ], 
  [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) ], 
  [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0 ] ]
gap> C:=GeneratorMatCode(G,GF(2));
a linear [6,3,1..3]2..3 code defined by generator matrix over GF(2)
gap> MinimumDistance(C);
2
gap> IsLinearCode(C);
true
gap> Elements(C);
[ [ 0 0 0 0 0 0 ], [ 0 0 1 0 1 1 ], [ 0 1 0 1 1 0 ], [ 0 1 1 1 0 1 ], 
  [ 1 0 0 1 1 0 ], [ 1 0 1 1 0 1 ], [ 1 1 0 0 0 0 ], [ 1 1 1 0 1 1 ] ]
gap> Dimension(C);
3

This is not 1 error correcting (try correcting [ 1 1 0 1 1 0 ]).

Here's another example,

gap> G := Z(2)*[ [1,0,0,1,1,0], [0,1,0,1,0,1], [0,0,1,0,1,1] ];
[ [ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) ], 
  [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0 ], 
  [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0 ] ]
gap> C:=GeneratorMatCode(G,GF(2));
a linear [6,3,1..3]2 code defined by generator matrix over GF(2)
gap> MinimumDistance(C);
3
gap> Dimension(C);
3
gap> Elements(C);
[ [ 0 0 0 0 0 0 ], [ 0 0 1 0 1 1 ], [ 0 1 0 1 0 1 ], [ 0 1 1 1 1 0 ], 
  [ 1 0 0 1 1 0 ], [ 1 0 1 1 0 1 ], [ 1 1 0 0 1 1 ], [ 1 1 1 0 0 0 ] ]

This is 1 error correcting.