Split Orthogonal Arrays and Maximum Independent ResilientSystems of Functions

• Authors: Vladimir I. Levenshtein
• Venue: Designs, Codes and Cryptography
• Year: 1997

A system of (Boolean) functions in n variablesis called randomized if the functions preserve the property oftheir variables to be independent and uniformly distributed randomvariables. Such a system is referred to as t-resilientif for any substitution of constants for any i variables,where 0 ≤ i ≤ t, the derived system of functionsin n-i variables will be also randomized. We investigatethe problem of finding the maximum number N(n,t,T)of functions in n variables of which any Tform a t-resilient system. This problem is reducedto the minimization of the size of certain combinatorial designs,which we call split orthogonal arrays. We extend some resultsof design and coding theory, in particular, a duality in boundingthe optimal sizes of codes and designs, in order to obtain upperand lower bounds on N(n,t,T). In some cases, thesebounds turn out to be very tight. In particular, for some infinitesubsequences of integers n they allow us to provethat N(n,3,3)=\frac{2^{n-2}}{n},N(n,3,5)=\sqrt{\frac{2^{n-1}}n},N(n,3,\frac n2-1)=n, N(n,\frac n2-1,3)=n, N(n,\frac n2-1,5)=\sqrt{2n}. We also find a connectionof the problem considered with the construction of unequal-error-protectioncodes and superimposed codes for multiple access in the Hammingchannel.