# Coding Strings by Pairs of Strings

## Abstract

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.

## References

1.
C. Hylten-Cavallius, On a combinatorical problem, Colloq. Math., 6 (1958), 59–65
2.
W. J. Paul, On heads versus tapes, Proc. 22nd Symposium on Foundations of Computer Science, 1981
3.
A. Zvonkin, L. Levin, The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms, Russian Math. Surveys, 6 (1970),
4.
P. Erdo˝s, A. Renyi, On the evolution of random graphsP. Erdo˝s, The art of counting, MIT Press, Cambridge, MA, 1973

## Information & Authors

### Information

#### Published In

SIAM Journal on Algebraic Discrete Methods
Pages: 445 - 461

#### History

Submitted: 22 September 1983
Accepted: 4 April 1984
Published online: 2 August 2006

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