The average dimension of the hull of cyclic codes. (Q1811111)

The average dimension \(E_q(n)\) of the hull of linear cyclic codes of a given length \(n\) over a given finite field \(F_q\) is studied, where the hull of a code is its intersection with its dual code. An explicit expression for \(E_q(n)\) is derived. Let \({\mathcal N}_q\) be the set of positive divisors of the integers of the form \(q^i+1\), \(i>0\). Then \(E_q(n)\) is zero if and only if \(n\in {\mathcal N}_q\), and it grows at the same rate as \(n\) when \(n\not\in {\mathcal N}_q\). Moreover, the asymptotic behavior of \(E_q(n)/n\) is studied, and the set of limits of all converging subsequences of the sequence \(E_q(n)/n\), \(n\geq 1\), \((n,q)=1\), is determined.