MathDB

Z_2

Source: Iranian National Olympiad (3rd Round) 2006

August 26, 2006

vectorgeometrygeometric transformationrotationalgebrapolynomialgroup theory

Problem Statement

n

is a natural number that

\frac{x^{n}+1}{x+1}

is irreducible over

\mathbb Z_{2}[x]

. Consider a vector in

\mathbb Z_{2}^{n}

that it has odd number of

1

's (as entries) and at least one of its entries are

0

. Prove that these vector and its translations are a basis for

\mathbb Z_{2}^{n}