The average dimension of the hull of cyclic codes

Authors:
Gintaras Skersys
Affiliations:
Matematikos ir Informatikos Fakultetas, Vilniaus Universitetas, Naugarduko 24, 2006 Vilnius, Lithuania
Venue:
Discrete Applied Mathematics - Special issue: International workshop on coding and cryptography (WCC 2001)
Year:
2003

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Abstract

We study Eq(n), the average dimension of the hull of error-correcting block cyclic codes of a given length n over a given finite field Fq, where the hull of a code is its intersection with its dual code. We derive an expression of Eq(n) which handles well. Using this expression, we prove that either Eq(n) is zero (if, and only if, n ∈ Nq), or it grows at the same rate as n, when n ∉ Nq, where Nq is the set of positive divisors of the integers of the form qi + 1, i 0. This permits us to show that, for almost all n, the hull of most cyclic codes of length n is "large". Moreover, we study the asymptotic behaviour of Eq(n)/n as n tends to infinity.