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Subsections
elt< C | a_1, ..., a_n> : Code, List -> ModTupRngElt
Given a code C which is defined as a subset of the
R-space R^((n)), and elements a_1, ..., a_n belonging
to R, construct the codeword (a_1, ..., a_n) of C.
It is checked that the vector (a_1, ..., a_n) is an element of C.
Given a code C which is defined as a subset of the R-space
V = R^((n)), and an element u belonging to V,
create the codeword of C corresponding to u. The function
will fail if u does not belong to C.
The zero word of the code C.
A random codeword of C.
Sum of the codewords u and v, where u and
v belong to the same linear code C.
Additive inverse of the codeword u belonging
to the linear code C.
Difference of the codewords u and v, where
u and v belong to the same linear code C.
Given an element a belonging to the field K,
and a codeword u belonging to the linear code
C, return the codeword a * u.
Given an element u over a field, not the zero element, belonging to
the linear code C, return (1/a) * u, where a is
the first non-zero component of u. If u is the zero vector,
it is returned as the value of this function. The net effect
is that Normalize(u) always returns a vector v in the
subspace generated by u, such that the first non-zero
component of v is the unit of K.
Given an [n, k] linear code C over a finite field
with parent vector space V,
and a vector w belonging to V, construct the syndrome of w
relative to the code C. This will be an element of the
syndrome space of C.
The Hamming distance between the codewords u and v,
where u and v belong to the same code C.
The Hamming weight of the codeword u, i.e., the number of non-zero
components of u.
The Lee weight of the codeword u.
We calculate all possible distances between code words of the
non-extended Golay code over GF(3), and show the correspondence with
all possible code word weights.
> C := GolayCode(GF(3),false);
> {Distance(v,w):v,w in C};
{ 0, 5, 6, 8, 9, 11 }
> {Weight(v):v in C};
{ 0, 5, 6, 8, 9, 11 }
InnerProduct(u, v) : ModTupRngElt, ModTupRngElt : -> RngElt
Inner product of the vectors u and v with respect to
the Euclidean norm, where u and v belong to the parent
vector space of the code C.
Given a word w belonging to the [n, k] code C, return its
support as a subset of the integer set { 1 .. n }. The support of w
consists of the coordinates at which w has non-zero entries.
Given an [n, k] linear code C and a codeword u of C return the
coordinates of u with respect to C. The coordinates of u are returned
as a sequence Q = [a_1, ..., a_k] of elements from the alphabet of C
so that u = a_1 * C.1 + ... + a_k * C.k.
Given a word w belonging to the code C, return the
ambient space V of C.
Given a vector u, return the vector obtained from u by
cyclically shifting its components to the right by k coordinate
positions.
Given a vector u, destructively rotate u by k coordinate positions.
Trace(u) : ModTupFldElt -> ModTupFldElt
Given a vector u with components in K, and a subfield S of K,
construct the vector with components in S obtained from u by taking
the trace of each component with respect to S. If S is omitted,
it is taken to be the prime field of K.
We create a specific code word in the length 5 even weight code,
after a failed attempt to create a code word of odd weight. We then
display its support, find its coordinates with respect to the basis
and then confirm it by way of re-construction.
> C := EvenWeightCode(5);
> C![1,1,0,1,0];
>> C![1,1,0,1,0];
^
Runtime error in '!': Result is not in the given structure
> c := C![1,1,0,1,1];
> c;
(1 1 0 1 1)
> Support(c);
{ 1, 2, 4, 5 }
> Coordinates(C,c);
[ 1, 1, 0, 1 ]
> C.1 + C.2 + C.4;
(1 1 0 1 1)
The function returns true if and only if the codewords u and v
are equal.
The function returns true if and only if the codewords u and v
are not equal.
The function returns true if and only if the codeword u is the
zero vector.
Given a codeword u belonging to the code C defined
over the ring R, return the i-th component of u
(as an element of R).
Given an element u belonging to a subcode C of the full R-space
V = R^n, a positive integer i, 1 <= i <= n, and an
element x of R, this function returns a vector in V
which is u with its i-th component redefined to be x.
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