Distribution of one-error linear complexity of binary sequences for arbitrary prime period (Q394411)

The linear complexity of a periodic binary sequence \(S\) is the length of the shortest linear recurrence relation (over the finite field \({\mathbb F}_2\)) that can generate \(S\). The \(k\)-error linear complexity of a periodic sequence is the smallest linear complexity that can be obtained by changing at most \(k\) terms per period. In this paper for any odd prime period \(N\), the exact formulas are given for the number \(\mathcal{N}(c^\prime,c)\) of \(N\)-periodic binary sequences with linear complexity \(c^\prime\) and \(1\)-error linear complexity \(c\), and for the number \(\mathcal{N}_1(c)\) of \(N\)-periodic binary sequences with \(1\)-error linear complexity \(c\).