/// <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);
}
/// <summary>
/// Construct the check matrix of a Goppa code in canonical form from the irreducible Goppa polynomial over the finite field <c>GF(2^m)</c>.
/// </summary>
///
/// <param name="Field">The finite field</param>
/// <param name="Gp">The irreducible Goppa polynomial</param>
///
/// <returns>The new GF2Matrix</returns>
public static GF2Matrix CreateCanonicalCheckMatrix(GF2mField Field, PolynomialGF2mSmallM Gp)
{
int m = Field.Degree;
int n = 1 << m;
int t = Gp.Degree;
// create matrix H over GF(2^m)
int[][] hArray = ArrayUtils.CreateJagged <int[][]>(t, n);
// create matrix YZ
int[][] yz = ArrayUtils.CreateJagged <int[][]>(t, n);
if (ParallelUtils.IsParallel)
{
Parallel.For(0, n, j =>
yz[0][j] = Field.Inverse(Gp.EvaluateAt(j)));
}
else
{
// here j is used as index and as element of field GF(2^m)
for (int j = 0; j < n; j++)
{
yz[0][j] = Field.Inverse(Gp.EvaluateAt(j));
}
}
for (int i = 1; i < t; i++)
{
// here j is used as index and as element of field GF(2^m)
if (ParallelUtils.IsParallel)
{
Parallel.For(0, n, j =>
{
yz[i][j] = Field.Multiply(yz[i - 1][j], j);
});
}
else
{
for (int j = 0; j < n; j++)
{
yz[i][j] = Field.Multiply(yz[i - 1][j], j);
}
}
}
// create matrix H = XYZ
for (int i = 0; i < t; i++)
{
if (ParallelUtils.IsParallel)
{
Parallel.For(0, n, j =>
{
for (int k = 0; k <= i; k++)
{
hArray[i][j] = Field.Add(hArray[i][j], Field.Multiply(yz[k][j], Gp.GetCoefficient(t + k - i)));
}
});
}
else
{
for (int j = 0; j < n; j++)
{
for (int k = 0; k <= i; k++)
{
hArray[i][j] = Field.Add(hArray[i][j], Field.Multiply(yz[k][j], Gp.GetCoefficient(t + k - i)));
}
}
}
}
// convert to matrix over GF(2)
int[][] result = ArrayUtils.CreateJagged <int[][]>(t * m, IntUtils.URShift((n + 31), 5));
if (ParallelUtils.IsParallel)
{
for (int j = 0; j < n; j++)
{
int q = IntUtils.URShift(j, 5);
int r = 1 << (j & 0x1f);
for (int i = 0; i < t; i++)
{
int e = hArray[i][j];
Parallel.For(0, m, u =>
{
int b = (IntUtils.URShift(e, u)) & 1;
if (b != 0)
{
int ind = (i + 1) * m - u - 1;
result[ind][q] ^= r;
}
});
}
}
}
else
{
for (int j = 0; j < n; j++)
{
int q = IntUtils.URShift(j, 5);
int r = 1 << (j & 0x1f);
for (int i = 0; i < t; i++)
{
int e = hArray[i][j];
for (int u = 0; u < m; u++)
{
int b = (IntUtils.URShift(e, u)) & 1;
if (b != 0)
{
int ind = (i + 1) * m - u - 1;
result[ind][q] ^= r;
}
}
}
}
}
return(new GF2Matrix(n, result));
}
/// <summary>
/// Construct the check matrix of a Goppa code in canonical form from the irreducible Goppa polynomial over the finite field <c>GF(2^m)</c>.
/// </summary>
///
/// <param name="Field">The finite field</param>
/// <param name="Gp">The irreducible Goppa polynomial</param>
///
/// <returns>The new GF2Matrix</returns>
public static GF2Matrix CreateCanonicalCheckMatrix(GF2mField Field, PolynomialGF2mSmallM Gp)
{
int m = Field.Degree;
int n = 1 << m;
int t = Gp.Degree;
// create matrix H over GF(2^m)
int[][] hArray = ArrayUtils.CreateJagged<int[][]>(t, n);
// create matrix YZ
int[][] yz = ArrayUtils.CreateJagged<int[][]>(t, n);
if (ParallelUtils.IsParallel)
{
Parallel.For(0, n, j =>
yz[0][j] = Field.Inverse(Gp.EvaluateAt(j)));
}
else
{
// here j is used as index and as element of field GF(2^m)
for (int j = 0; j < n; j++)
yz[0][j] = Field.Inverse(Gp.EvaluateAt(j));
}
for (int i = 1; i < t; i++)
{
// here j is used as index and as element of field GF(2^m)
if (ParallelUtils.IsParallel)
{
Parallel.For(0, n, j =>
{
yz[i][j] = Field.Multiply(yz[i - 1][j], j);
});
}
else
{
for (int j = 0; j < n; j++)
yz[i][j] = Field.Multiply(yz[i - 1][j], j);
}
}
// create matrix H = XYZ
for (int i = 0; i < t; i++)
{
if (ParallelUtils.IsParallel)
{
Parallel.For(0, n, j =>
{
for (int k = 0; k <= i; k++)
hArray[i][j] = Field.Add(hArray[i][j], Field.Multiply(yz[k][j], Gp.GetCoefficient(t + k - i)));
});
}
else
{
for (int j = 0; j < n; j++)
{
for (int k = 0; k <= i; k++)
hArray[i][j] = Field.Add(hArray[i][j], Field.Multiply(yz[k][j], Gp.GetCoefficient(t + k - i)));
}
}
}
// convert to matrix over GF(2)
int[][] result = ArrayUtils.CreateJagged<int[][]>(t * m, IntUtils.URShift((n + 31), 5));
if (ParallelUtils.IsParallel)
{
for (int j = 0; j < n; j++)
{
int q = IntUtils.URShift(j, 5);
int r = 1 << (j & 0x1f);
for (int i = 0; i < t; i++)
{
int e = hArray[i][j];
Parallel.For(0, m, u =>
{
int b = (IntUtils.URShift(e, u)) & 1;
if (b != 0)
{
int ind = (i + 1) * m - u - 1;
result[ind][q] ^= r;
}
});
}
}
}
else
{
for (int j = 0; j < n; j++)
{
int q = IntUtils.URShift(j, 5);
int r = 1 << (j & 0x1f);
for (int i = 0; i < t; i++)
{
int e = hArray[i][j];
for (int u = 0; u < m; u++)
{
int b = (IntUtils.URShift(e, u)) & 1;
if (b != 0)
{
int ind = (i + 1) * m - u - 1;
result[ind][q] ^= r;
}
}
}
}
}
return new GF2Matrix(n, result);
}
