galois.ReedSolomon.roots

property galois.ReedSolomon.roots : FieldArray

The $d - 1$ roots of the generator polynomial $g (x)$ .

These are consecutive powers of $α^{c}$ , specifically $α^{c}, \dots, α^{c + d - 2}$ .

Examples¶

Construct a narrow-sense $RS (15, 9)$ code over $GF (2^{4})$ with first consecutive root $α$ .

In [1]: rs = galois.ReedSolomon(15, 9); rs
Out[1]: <Reed-Solomon Code: [15, 9, 7] over GF(2^4)>

In [2]: rs.roots
Out[2]: GF([ 2,  4,  8,  3,  6, 12], order=2^4)

In [3]: rs.generator_poly
Out[3]: Poly(x^6 + 7x^5 + 9x^4 + 3x^3 + 12x^2 + 10x + 12, GF(2^4))

# Evaluate the generator polynomial at its roots in GF(q)
In [4]: rs.generator_poly(rs.roots)
Out[4]: GF([0, 0, 0, 0, 0, 0], order=2^4)

Construct a non-narrow-sense $RS (15, 9)$ code over $GF (2^{4})$ with first consecutive root $α^{3}$ .

In [5]: rs = galois.ReedSolomon(15, 9, c=3); rs
Out[5]: <Reed-Solomon Code: [15, 9, 7] over GF(2^4)>

In [6]: rs.roots
Out[6]: GF([ 8,  3,  6, 12, 11,  5], order=2^4)

In [7]: rs.generator_poly
Out[7]: Poly(x^6 + 15x^5 + 8x^4 + 7x^3 + 9x^2 + 3x + 8, GF(2^4))

# Evaluate the generator polynomial at its roots in GF(q)
In [8]: rs.generator_poly(rs.roots)
Out[8]: GF([0, 0, 0, 0, 0, 0], order=2^4)