guava - Gap

Chapter 2
A First Tutorial in GUAVA
An error-correcting code is essentially just a subset of the set of all possible messages of a given length
over some finite ”alphabet.”
In algebraic coding theory, the ”alphabet” is usually some finite field (very often GF(2)) and
frequently the error-correcting code is chosen to be a vector subspace of the space of all row vectors
of some fixed length n. Such codes are known as Linear Codes, but, however a code is defined the
point is to have a collection of ”codewords” that are said to be ”in the code” and any other word (row
vectors that are not ”in the code”) will be assumed to be a codeword that has been mangled by the
addition of noise.
When a message is received that is not a codeword, we ask ourselves the question ”Which codeword
is closest to this message I’ve received?” In other words we make the presumption that the
received message is actually a codeword that has been changed in a relatively small number of positions
– and we put them back the way they were supposed to be!
That process is called ”decoding.” Developing codes that have efficient decoding algorithms is
one of the central problems of algebraic coding theory.
2.1 Working with codewords
So let’s play around a bit.
Start GAP in a terminal window, then issue the command
Example
gap> LoadPackage("guava");
GUAVA can construct codewords in a variety of ways. One of the most typical cases is for a
codeword to consist of binary digits. In that case we say that ”the code is over GF(2)” and codewords
can be constructed as follows:
Example
gap> c1:=Codeword("101010101");
[ 1 0 1 0 1 0 1 0 1 ]
gap> v:=Z(2)*[1,1,1,1,1,1,1,1,1];
[ Z(2)ˆ0, Z(2)ˆ0, Z(2)ˆ0, Z(2)ˆ0, Z(2)ˆ0, Z(2)ˆ0, Z(2)ˆ0, Z(2)ˆ0, Z(2)ˆ0 ]
gap> c2:=Codeword(v);
[ 1 1 1 1 1 1 1 1 1 ]
gap> c3:=c1+c2;
[ 0 1 0 1 0 1 0 1 0 ]
gap> Weight(c1);
14