393 research outputs found
A Lower Bound on the Constant in the Fourier Min-Entropy/Influence Conjecture
We describe a new construction of Boolean functions. A specific instance of
our construction provides a 30-variable Boolean function having
min-entropy/influence ratio to be which is presently
the highest known value of this ratio that is achieved by any Boolean function.
Correspondingly, is also presently the best known lower bound on the
universal constant of the Fourier min-entropy/influence conjecture
On Using Proportional Representation Methods as Alternatives to Pro-Rata Based Order Matching Algorithms in Stock Exchanges
The main observation of this short note is that methods for determining
proportional representation in electoral systems may be suitable as
alternatives to the pro-rata order matching algorithm used in stock exchanges.
Our simulation studies provide strong evidence that the Jefferson/D'Hondt and
the Webster/Saint-Lagu\"{e} proportional representation methods provide order
allocations which are closer to proportionality than the order allocations
obtained from the pro-rata algorithm
A Direct Construction of Intergroup Complementary Code Set for CDMA
A collection of mutually orthogonal complementary codes (CCs) is said to be complete complementary codes (CCCs) where the number of CCs are equal to the number of constituent sequences in each CC. Intergroup complementary (IGC) code set is a collection of multiple disjoint code groups with the following correlation properties: (1) inside the zero-correlation zone (ZCZ), the aperiodic autocorrelation function (AACF) of any IGC code is zero for all nonzero time shifts; (2) the aperiodic cross-correlation function (ACCF), of two distinct IGC codes, is zero for all time shifts inside the ZCZ when they are taken from the same code groups; and (3) the ACCF, for two IGC codes from two different code groups, is zero everywhere. IGC code set has a larger set size than CCC, and both can be applicable in multicarrier code-division multiple access (CDMA). In this chapter, we present a direct construction of IGC code set by using second-order generalized Boolean functions (GBFs), and our IGC code set can support interference-free code-division multiplexing. We also relate our construction with a graph where the ZCZ width depends on the number of isolated vertices present in a graph after the deletion of some vertices. Here, the construction that we propose can generate IGC code set with more flexible parameters
Influence of a Set of Variables on a Boolean Function
The influence of a variable is an important concept in the analysis of
Boolean functions. The more general notion of influence of a set of variables
on a Boolean function has four separate definitions in the literature. In the
present work, we introduce a new definition of influence of a set of variables
which is based on the auto-correlation function and develop its basic theory.
Among the new results that we obtain are generalisations of the Poincar\'{e}
inequality and the edge expansion property of the influence of a single
variable. Further, we obtain new characterisations of resilient and bent
functions using the notion of influence. We show that the previous definition
of influence due to Fischer et al. (2002) and Blais (2009) is half the value of
the auto-correlation based influence that we introduce. Regarding the other
prior notions of influence, we make a detailed study of these and show that
each of these definitions do not satisfy one or more desirable properties that
a notion of influence may be expected to satisfy
Reducing Communication Overhead of the Subset Difference Scheme
In Broadcast Encryption (BE) systems like Pay-TV, AACS, online content sharing and broadcasting, reducing the header length (communication overhead per session) is of practical interest. The Subset Difference (SD) scheme due to Naor-Naor-Lotspiech (NNL) is the most popularly used BE scheme. We introduce the (a, b, γ) augmented binary tree subset difference ( (a, b, γ) -ABTSD) scheme which is a generalization of the NNL-SD scheme. By varying the parameters (a, b, γ) , it is possible to obtain O(n log n) different schemes. The average header length achieved by the new schemes is smaller than all known schemes having the same decryption time as that of the NNL-SD scheme and achieving non-trivial trade-offs between the user storage and the header size. The amount of key material that a user is required to store increases. For the earlier mentioned applications, reducing header size and achieving fast decryption is perhaps more of a concern than the user storage
- …