galois.BCH.alpha

property galois.BCH.alpha : FieldArray

A primitive $n$ -th root of unity $α$ in $GF (q^{m})$ whose consecutive powers $α^{c}, \dots, α^{c + d - 2}$ are roots of the generator polynomial $g (x)$ in $GF (q^{m})$ .

Examples

Construct a binary primitive $BCH (15, 7)$ code.

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

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

In [3]: bch.roots[0] == bch.alpha ** bch.c
Out[3]: True

In [4]: bch.alpha.multiplicative_order() == bch.n
Out[4]: True

Construct a non-primitive $BCH (13, 7)$ code over $GF (3)$ .

In [5]: bch = galois.BCH(13, 7, field=galois.GF(3)); bch
Out[5]: <BCH Code: [13, 7, 4] over GF(3)>

In [6]: bch.alpha
Out[6]: GF(9, order=3^3)

In [7]: bch.roots[0] == bch.alpha ** bch.c
Out[7]: True

In [8]: bch.alpha.multiplicative_order() == bch.n
Out[8]: True