Subsections

• Construction of a Codeword
• Elementary Operations on Codewords and Vectors
• Accessing Components of a Codeword

elt< C | a_1, ..., a_n> : Code, List -> ModTupFldElt

Given a code C which is defined as a subset of the vector space K^((n)), and elements a_1, ..., a_n belonging to K, construct the codeword (a_1, ..., a_n) of C. It is checked that the vector (a_1, ..., a_n) is an element of C.

C ! u : Code, ModTupFldElt -> ModTupFldElt

Given a code C which is defined as a subset of the vector space V = K^((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.

C ! 0 : Code, RngIntElt -> ModTupFldElt

The zero word of the code 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.

u - v : ModTupFldElt, ModTupFldElt -> ModTupFldElt

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 [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.

The Hamming distance between the codewords u and v, where u and v belong to the same code C.

InnerProduct(u, v) : ModTupFldElt, ModTupFldElt : -> 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.

The Lee weight of the codeword u.

Given an element u, 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 a vector u, return the vector obtained from u by rotating by k coordinate positions.

Rotate(~u, k) : ModTupFldElt, RngIntElt ->

Given a vector u, destructively rotate u by k coordinate positions.

Given an [n, k] linear code C 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.

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.

The Hamming weight of the codeword u, i.e. the number of non-zero components of u.

> C := GolayCode(GF(3), false);
> { Distance(v, w): v, w in C };
{ 0, 5, 6, 8, 9, 11 }

Given a codeword u belonging to the code C defined over the field K, return the i-th component of u (as an element of the field K).

u[i] := x;

Given an element u belonging to a subcode C of the full vector space V = K^n, a positive integer i, 1 <= i <= n, and an element x of the field K, redefine the i-th component of u to be x.