Skip to main content

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 \(\left (x^{\frac {a}{3}}\right)^{30}+\left (x^{\frac {a}{3} }\right)^{9}+1\)