Massimiliano Sala edited bits3.tex  over 7 years ago

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producing $n$-bit vectors and such that any initial state is got again after  a finite number of iterations.\\  Fact (\ref{fatc-b}) is more difficult to generalize. If we want an LFSR that can start from any initial nonzero vector and obtain all other nonzero vectors, then we must use very special polynomials, called \emph{primitive} polynomials.  In other words %  \begin{Definition}\label{Primitive}  We call primitive a any  polynomial $f$ $f\in \FF_2[x]$  of degree $n$ that, used as feedback polynomials of a LFSR, can generate all the non-zero  $n$-tuples of bits with at least one nonzero entry,i.e. bits,i.e.  all the $n$-tuples but $(0,0,0,...,0,0)$. $(0,0,0,...,0,0)$, using as initial  state any of these.  \end{Definition}