# Non linear covering codes : A few results and conjectures

• Gérard D. Cohen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 356)

## Abstract

Denote by K(n,t) the minimum number of elements in a covering of IF 2 n with Hamming spheres of radius t, and by μn(t) the density of such a covering :
$$\mu _n (t): = 2^{ - n} K(n,t) \mathop \Sigma \limits_{i = 0}^t ( \mathop {}\limits_i^n ).$$
For fixed t, μn (t) is upperbounded by a real number c(t) independent of n. For t=1, one can take c(1)=1.5. We conjecture : Conjecture 1.$$\mathop {lim}\limits_{n \to \infty } \mu _n (1) = 1.$$ μn(1)=1. Another conjecture is : Conjecture 2. K(n+2,t+1)≤K(n,t). We prove this for t=1 and discuss a possible way of proving it for higher t, by extensions of the concept of normality.

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