### Bounds for the Multicovering Radii of Reed-Muller
Codes with Applications to Stream Ciphers

Designs, Codes, and Cryptography ** 23 ** (2001) 131-145.
Authors:

Andrew Klapper, 779A Anderson Hall, Dept. of Computer Science,
University of Kentucky, Lexington, KY, 40506-0046, klapper at cs.uky.edu.

Iiro Honkala, University of Turku

**Abstract**
The multicovering radii of a code are recent generalizations of
the covering radius of a code. For positive m, the m-covering
radius of C is the least radius t such that every m-tuple of
vectors is contained in at least one ball of radius t centered at
some codeword. In this paper upper bounds are found for
the multicovering radii of first order Reed-Muller codes. These bounds
generalize the well-known Norse bounds for the classical covering radii
of first order Reed-Muller codes. They are exact
in some cases. These bounds are then used to prove the existence of
secure families of keystreams against a general class of cryptanalytic
attacks. This solves the open question that gave rise to the study of
multicovering radii of codes.

**Index Terms --**
Error correcting code, stream cipher, covering
radius, Reed-Muller code.