Weight Enumerators

CompleteWeightEnumerator(C): Code -> RngMPolElt

Let \(C\) be a code over a finite ring \(R\) of cardinality \(q\), and suppose that the elements of \(R\) are ordered in some way. Then for a codeword \(v \in C\) and the \(i\)-th element \(a \in R\), let \(s_i(v)\) denote the number of components of \(v\) equal to \(a\).

This function returns the complete weight enumerator \({\cal W}_C(X_0, X_1, \ldots, X_{q-1})\) of \(C\), which is defined by:

\[{\cal W}_C(X_0, X_1, \ldots, X_{q-1}) = \sum_{v \in C}{{X_0}^{s_0(v)}{X_1}^{s_1(v)}\cdots{X_{q-1}}^{s_{q-1}(v)}}.\]

See [Wan, 1997, p. 9] for more information. The result will lie in a global multivariate polynomial ring over \({\mathbb{Z}}\) with q variables. The angle-bracket notation may be used to assign names to the indeterminates.

WeightEnumerator(C): Code -> RngMPolElt

HammingWeightEnumerator(C): Code -> RngMPolElt

Suppose \(C\) is a code over some finite ring \(R\). This function returns the Hamming weight enumerator \(\hbox{Ham}_C(X, Y)\) of \(C\), which is defined by:

\[\hbox{Ham}_C(X, Y) = \sum_{v \in C}{X^{n-w_H(v)}Y^{w_H(v)}},\]

where \(w_H(v)\) is the Hamming weight function. The result will lie in a global multivariate polynomial ring over \({\mathbb{Z}}\) with two variables. The angle-bracket notation may be used to assign names to the indeterminates.

Example: WeightEnum Galois Rings (ex-1d525d)

We compute the complete weight enumerator of a cyclic code over the Galois ring GR\((4, 2)\).

> R<w> := GR(4,2);
> P<x> := PolynomialRing(R);
> L := CyclotomicFactors(R, 3);
> g := L[1];
> g;
x + 3
> C := CyclicCode(3, g);
> C;
(3, 256, 2) Cyclic Code over GaloisRing(2, 2, 2)
Generator matrix:
[1 0 3]
[0 1 3]
> CWE<[X]> := CompleteWeightEnumerator(C);
> CWE;
X[1]^3 + 6*X[1]*X[2]*X[4] + 3*X[1]*X[3]^2 + 6*X[1]*X[5]*X[13] +
    6*X[1]*X[6]*X[16] + 6*X[1]*X[7]*X[15] + 6*X[1]*X[8]*X[14] +
    3*X[1]*X[9]^2 + 6*X[1]*X[10]*X[12] + 3*X[1]*X[11]^2 +
    3*X[2]^2*X[3] + 6*X[2]*X[5]*X[16] + 6*X[2]*X[6]*X[15] +
    6*X[2]*X[7]*X[14] + 6*X[2]*X[8]*X[13] + 6*X[2]*X[9]*X[12] +
    6*X[2]*X[10]*X[11] + 3*X[3]*X[4]^2 + 6*X[3]*X[5]*X[15] +
    6*X[3]*X[6]*X[14] + 6*X[3]*X[7]*X[13] + 6*X[3]*X[8]*X[16] +
    6*X[3]*X[9]*X[11] + 3*X[3]*X[10]^2 + 3*X[3]*X[12]^2 +
    6*X[4]*X[5]*X[14] + 6*X[4]*X[6]*X[13] + 6*X[4]*X[7]*X[16] +
    6*X[4]*X[8]*X[15] + 6*X[4]*X[9]*X[10] + 6*X[4]*X[11]*X[12] +
    3*X[5]^2*X[9] + 6*X[5]*X[6]*X[12] + 6*X[5]*X[7]*X[11] +
    6*X[5]*X[8]*X[10] + 3*X[6]^2*X[11] + 6*X[6]*X[7]*X[10] +
    6*X[6]*X[8]*X[9] + 3*X[7]^2*X[9] + 6*X[7]*X[8]*X[12] +
    3*X[8]^2*X[11] + 3*X[9]*X[13]^2 + 6*X[9]*X[14]*X[16] +
    3*X[9]*X[15]^2 + 6*X[10]*X[13]*X[16] + 6*X[10]*X[14]*X[15] +
    6*X[11]*X[13]*X[15] + 3*X[11]*X[14]^2 + 3*X[11]*X[16]^2 +
    6*X[12]*X[13]*X[14] + 6*X[12]*X[15]*X[16]

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