An error correcting code of length * n * is a set * C * of vectors in
* n *-dimensional space over * GF(2) *. The ** covering radius** of
a code * C * is the smallest integer * r * such that every vector in
* GF(2)^n * is within distance * r * of at least one code word in
*C*. This notion plays a crucial role in coding theory
and has been the subject of hundreds of papers over the past two
decades (The notion of covering radius also arises in horserace
betting pools. Some of the earliest results in the area were found by
nonmathematicians and published in betting pool magazines!.) A critical step
in the proof of one of the existence theorems in my paper *On the Existence
of Secure Keystream Generators* requires the existence of a code
with small covering radius each of whose code words can be generated by a
fast feedback register. The Reed-Muller codes satisfy these requirements.
This gives a result that says there exists a family of
sequences that resists all attacks of the type studied, in the sense that each
attack is wrong on close to half the bits for infinitely many sequences in
the family.

A stronger result would say that every attack fails on all but finitely
many of the sequences. It would then be possible for a cryptographer to
choose a sequence that resists any given finite set of attacks. To prove such
a result one needs a code with small {\em multicovering radius}. If *m
* is a positive integer, then the * m*-covering radius of * C * is
the smallest integer * r * such that for every set *{v1,...,vm}* in
*GF(2)^n *, there is a vector * c* in C such that the
distance from every * vi * to * c * is at most * r *.
Surprisingly, this is a new concept, previously unstudied by coding theorists.
In the paper *The Multicovering Radii of Codes* I have initiated its
study. This paper describes the basic properties of multicovering radii and gives
several lower and upper bounds. A natural question is, for a given * r *,
* m *, and * n *, what is the smallest code whose * m *-covering
radius is at most * r *? As it turns out, there may be no such code. For
example, if * m\ge 2 *, it is necessary that * r* be at least * n/2
*. In fact, the minimal * r * for which such a code exists is the *
m*-covering radius of * C=GF(2)^n *. I have derived tight asymptotic
bounds on these radii. I also derived tight asymptotic bounds on the *
m*-covering radii of Hamming codes.

- Multicovering Radii: (abstract)
**The Multicovering Radii of Codes**. - Secure Keystream Generators: (abstract)
**On the Existence of Secure Keystream Generators**.