On the limit of maximal density of sequences with a perfect linear complexity profile (Q676731)

For a given (finite or infinite) binary ultimately periodic sequence \((s_1,s_2,\dots)\) its linear complexity is defined as the smallest number \(k\) such that \(s_{k+i}=c_1s_i+\cdots+c_ks_{i+k-1}\) over GF(2), for all \(i\geq1\), where \(c_1,\dots,c_k\) are some binary integers. The linear complexity profile of \((s_1,s_2,\dots)\) is the corresponding sequence \(L(1),L(2),\dots\) of values of the linear complexity of the sequences: \((s_1)\), \((s_1,s_2)\), \(\dots\). If \(L(n)=\lfloor{n+1\over2}\rfloor\) for all \(n\geq1\) then \(L\) is called the perfect linear complexity profile. Massey and Wang have shown that a binary sequence \((s_1,\dots,s_n)\) has the perfect linear complexity profile if and only if \(s_1=1\) and \(s_{2i+1}=s_{2i}+s_i\), for \(i\geq1\). The paper contains a proof that the limit of the maximum density (meant as \(\sum_{i=1}^ns_i/n\)) over all such sequences is equal to 2/3, as \(n\to\infty\).