/// <summary>
/// Find an error vector <c>E</c> over <c>GF(2)</c> from an input syndrome <c>S</c> over <c>GF(2^M)</c>
/// </summary>
///
/// <param name="SyndVec">The syndrome</param>
/// <param name="Field">The finite field</param>
/// <param name="Gp">The irreducible Goppa polynomial</param>
/// <param name="SqRootMatrix">The matrix for computing square roots in <c>(GF(2M))<sup>T</sup></c></param>
///
/// <returns>The error vector</returns>
public static GF2Vector SyndromeDecode(GF2Vector SyndVec, GF2mField Field, PolynomialGF2mSmallM Gp, PolynomialGF2mSmallM[] SqRootMatrix)
{
int n = 1 << Field.Degree;
// the error vector
GF2Vector errors = new GF2Vector(n);
// if the syndrome vector is zero, the error vector is also zero
if (!SyndVec.IsZero())
{
// convert syndrome vector to polynomial over GF(2^m)
PolynomialGF2mSmallM syndrome = new PolynomialGF2mSmallM(SyndVec.ToExtensionFieldVector(Field));
// compute T = syndrome^-1 mod gp
PolynomialGF2mSmallM t = syndrome.ModInverse(Gp);
// compute tau = sqRoot(T + X) mod gp
PolynomialGF2mSmallM tau = t.AddMonomial(1);
tau = tau.ModSquareRootMatrix(SqRootMatrix);
// compute polynomials a and b satisfying a + b*tau = 0 mod gp
PolynomialGF2mSmallM[] ab = tau.ModPolynomialToFracton(Gp);
// compute the polynomial a^2 + X*b^2
PolynomialGF2mSmallM a2 = ab[0].Multiply(ab[0]);
PolynomialGF2mSmallM b2 = ab[1].Multiply(ab[1]);
PolynomialGF2mSmallM xb2 = b2.MultWithMonomial(1);
PolynomialGF2mSmallM a2plusXb2 = a2.Add(xb2);
// normalize a^2 + X*b^2 to obtain the error locator polynomial
int headCoeff = a2plusXb2.Head;
int invHeadCoeff = Field.Inverse(headCoeff);
PolynomialGF2mSmallM elp = a2plusXb2.MultWithElement(invHeadCoeff);
// for all elements i of GF(2^m)
for (int i = 0; i < n; i++)
{
// evaluate the error locator polynomial at i
int z = elp.EvaluateAt(i);
// if polynomial evaluates to zero
if (z == 0)
{
// set the i-th coefficient of the error vector
errors.SetBit(i);
}
}
}
return(errors);
}
