property galois.ReedSolomon.roots : FieldArray

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

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

Examples

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

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 $$\textrm{RS}(15, 9)$$ code over $$\mathrm{GF}(2^4)$$ with first consecutive root $$\alpha^3$$.

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)