property galois.BCH.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 binary narrow-sense $$\textrm{BCH}(15, 7)$$ code with first consecutive root $$\alpha$$.

In [1]: bch = galois.BCH(15, 7); bch
Out[1]: <BCH Code: [15, 7, 5] over GF(2)>

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

In [3]: bch.generator_poly
Out[3]: Poly(x^8 + x^7 + x^6 + x^4 + 1, GF(2))

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


Construct a binary non-narrow-sense $$\textrm{BCH}(15, 7)$$ code with first consecutive root $$\alpha^3$$. Notice the design distance of this code is only 3 and it only has 2 roots in $$\mathrm{GF}(2^4)$$.

In [5]: bch = galois.BCH(15, 7, c=3); bch
Out[5]: <BCH Code: [15, 7, 3] over GF(2)>

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

In [7]: bch.generator_poly
Out[7]: Poly(x^8 + x^7 + x^6 + x^4 + 1, GF(2))

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