Robust parent-identifying codes and combinatorial arrays
Alexander Barg and Grigory Kabatiansky
Abstract
An -word over a finite alphabet of cardinality is called a descendant of a set of words if for all A code is said to have the -IPP property if for any -word that is a descendant of at most parents belonging to the code it is possible to identify at least one of them. From earlier works it is known that -IPP codes of positive rate exist if and only if .
We introduce a robust version of IPP codes which allows {unconditional} identification of parents even if some of the coordinates in can break away from the descent rule, i.e., can take arbitrary values from the alphabet, or become completely unreadable. We show existence of robust -IPP codes
for all and some positive proportion of such coordinates.
The proofs involve relations between IPP codes and combinatorial arrays with separating properties such as perfect hash functions and hash codes, partially hashing families and separating codes.
For we find the exact proportion of mutant coordinates (for several error scenarios) that permits unconditional identification
of parents.