Coding Strings by Pairs of Strings

Let X, $Y \subset \{ 0,1 \}^* $. We say Y codes X if every $x \in X$ can be obtained by applying a short program to some $y \in Y$. We are interested in sets Y that code X robustly in the sense that even if we delete an arbitrary subset $Y' \subset Y$ of size k, say, the remaining set of strings $Y\backslash Y'$ still codes X. In general, this can be achieved only by making in some sense more than k copies of each $x \in X$ and distributing these copies on different strings Y. Thus if the strings in X and Y have the same length, then $# \,Y\geqq ( k + 1 )# X$ .

If we allow coding of X by Y in a way that every $x \in X$ is obtained from strings x, $z \in Y$ by application of a short program, then we can do better.

Let $Y = \{ \oplus _{x \in S} x |S \subset X \}$ where $ \oplus $ denotes bitwise sum mod 2. Then $# Y = 2^{ # X} $. Yet Y codes X robustly for $k = 2^{ # X - 1} - 1$. This paper explores the limitations of coding schemes of this nature.