Table 2 Irreducible polynomials \(p\left (x^{\frac {a}{b}}\right)\) against primitive polynomials p(x a)
From: An association between primitive and non-primitive BCH codes using monoid rings
deg | p(x a) | \(p\left (x^{\frac {a}{b}}\right)\) |
|---|---|---|
2 | (x a)2+(x a)+1 | \(\left (x^{\frac {a}{3}}\right)^{6}+\left (x^{\frac {a}{3}}\right)^{3}+1\) |
3 | (x a)3+(x a)+1 | \(\left (x^{\frac {a}{7}}\right)^{21}+\left (x^{\frac {a}{7}}\right)^{7}+1\) |
4 | (x a)4+(x a)+1 | \(\left (x^{\frac {a}{3}}\right)^{12}+\left (x^{\frac {a}{3}}\right)^{3}+1,\left (x^{\frac {a}{5}}\right)^{20}+\left (x^{\frac {a}{5}}\right)^{5}+1\) |
6 | (x a)6+(x a)+1 | \(\left (x^{\frac {a}{3}}\right)^{18}+\left (x^{\frac {a}{3}}\right)^{3}+1,\left (x^{\frac {a}{7}}\right)^{42}+\left (x^{\frac {a}{7}}\right)^{7}+1\) |
8 | (x a)8+(x a)4+(x a)3 | \(\left (x^{\frac {a}{3}}\right)^{24}+\left (x^{\frac {a}{3}}\right)^{12}+\left (x^{\frac {a}{3}}\right)^{9}+\left (x^{\frac {a}{3} }\right)^{6}+1,\) |
+(x a)2+1 | ||
\(\left (x^{\frac {a}{5}}\right)^{40}+\left (x^{\frac {a}{5}}\right)^{20}+\left (x^{\frac {a}{5} }\right)^{15}+\left (x^{\frac {a}{5}}\right)^{10}+1\) | ||
9 | (x a)9+(x a)4+1 | \(\left (x^{\frac {a}{7}}\right)^{63}+\left (x^{\frac {a}{7} }\right)^{28}+1\) |
10 | (x a)10+(x a)3+1 | \(\left (x^{\frac {a}{3}}\right)^{30}+\left (x^{\frac {a}{3} }\right)^{9}+1\) |
â‹® | â‹® | â‹® |