(n,k,t))-covering systems and error-trapping decoding (Corresp.)

The technique of error-trapping decoding for algebraic codes is studied in combinatorial terms of covering systems. Let
n, k
, and
t
be positive integers such that
n \geq k \geq t > 0
. An
(n, k,t)
-covering system is a pair
(X, \beta)
, where
X
is a set of size
n
and
\beta
is a collection of subsets of
X
, each of size
k
, such that for all
T \subseteq X
of size
t
, there exists at least one
B \in \beta
with
T\subseteq B
. Let
b(n, k, t)
denote the smallest size of
\beta
, such that
(X, \beta)
is an
(n, k, t)
-covering system. It is shown that the complexity of an error-trapping decoding technique is bounded by
b(n, k, t)
from below. Two new methods for constructing small
(n, k, t)
-covering systems, the algorithmic method and the difference family method, are given.