Algebraic and Stochastic Coding Theory

C

Tables in Finite Fields

C.1    Table of Irreducible Polynomials in F₂[x]

The following definitions and explanations are necessary for understanding the usefulness of Table C.1.

1.  Given a polynomial $p(x)={\displaystyle \sum _{k=0}^n{a}_k{x}^k\in F_2[x]}$, the reciprocal polynomial p*(x) is defined by $p^{*}(x)={\displaystyle \sum _{k=0}^n{a}_k{x}^{n-k}}$. For example, $p(x)=a+bx+c{x}^2\leftrightarrow (a,b,c)\Rightarrow p^{*}(x)\leftrightarrow {(a,b,c)}^{*}=(c,b,a)\leftrightarrow c+bx+a{x}^2$.

2.  If is irreducible (primitive) and has a zero α, then p*(x) is irreducible (primitive) and has zero α⁻¹. It means that in Table C.1 only one out of the pair p(x) and p*(x) is listed; that is, the sequence of coefficients may be read either from the left or from the right to obtain an irreducible polynomial.

3.  For each degree, let α be a zero of the first listed polynomial. The entry following ...