1,987 research outputs found
Asymptotic enumeration of correlation-immune boolean functions
A boolean function of boolean variables is {correlation-immune} of order
if the function value is uncorrelated with the values of any of the
arguments. Such functions are of considerable interest due to their
cryptographic properties, and are also related to the orthogonal arrays of
statistics and the balanced hypercube colourings of combinatorics. The {weight}
of a boolean function is the number of argument values that produce a function
value of 1. If this is exactly half the argument values, that is,
values, a correlation-immune function is called {resilient}.
An asymptotic estimate of the number of -variable
correlation-immune boolean functions of order was obtained in 1992 by
Denisov for constant . Denisov repudiated that estimate in 2000, but we will
show that the repudiation was a mistake.
The main contribution of this paper is an asymptotic estimate of
which holds if increases with within generous limits and specialises to
functions with a given weight, including the resilient functions. In the case
of , our estimates are valid for all weights.Comment: 18 page
On unbalanced Boolean functions with best correlation immunity
It is known that the order of correlation immunity of a nonconstant
unbalanced Boolean function in variables cannot exceed ; moreover,
it is if and only if the function corresponds to an equitable
-partition of the -cube with an eigenvalue of the quotient matrix.
The known series of such functions have proportion , , or of
the number of ones and zeros. We prove that if a nonconstant unbalanced Boolean
function attains the correlation-immunity bound and has ratio of the
number of ones and zeros, then is divisible by . In particular, this
proves the nonexistence of equitable partitions for an infinite series of
putative quotient matrices. We also establish that there are exactly
equivalence classes of the equitable partitions of the -cube with quotient
matrix and classes, with . These
parameters correspond to the Boolean functions in variables with
correlation immunity and proportion and , respectively (the case
remains unsolved). This also implies the characterization of the
orthogonal arrays OA and OA.Comment: v3: final; title changed; revised; OA(512,11,2,6) discusse
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 page
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