6,346 research outputs found
Linear Codes from Some 2-Designs
A classical method of constructing a linear code over \gf(q) with a
-design is to use the incidence matrix of the -design as a generator
matrix over \gf(q) of the code. This approach has been extensively
investigated in the literature. In this paper, a different method of
constructing linear codes using specific classes of -designs is studied, and
linear codes with a few weights are obtained from almost difference sets,
difference sets, and a type of -designs associated to semibent functions.
Two families of the codes obtained in this paper are optimal. The linear codes
presented in this paper have applications in secret sharing and authentication
schemes, in addition to their applications in consumer electronics,
communication and data storage systems. A coding-theory approach to the
characterisation of highly nonlinear Boolean functions is presented
On a conjecture of Helleseth
We are concern about a conjecture proposed in the middle of the seventies by
Hellesseth in the framework of maximal sequences and theirs cross-correlations.
The conjecture claims the existence of a zero outphase Fourier coefficient. We
give some divisibility properties in this direction
Quantum algorithms for highly non-linear Boolean functions
Attempts to separate the power of classical and quantum models of computation
have a long history. The ultimate goal is to find exponential separations for
computational problems. However, such separations do not come a dime a dozen:
while there were some early successes in the form of hidden subgroup problems
for abelian groups--which generalize Shor's factoring algorithm perhaps most
faithfully--only for a handful of non-abelian groups efficient quantum
algorithms were found. Recently, problems have gotten increased attention that
seek to identify hidden sub-structures of other combinatorial and algebraic
objects besides groups. In this paper we provide new examples for exponential
separations by considering hidden shift problems that are defined for several
classes of highly non-linear Boolean functions. These so-called bent functions
arise in cryptography, where their property of having perfectly flat Fourier
spectra on the Boolean hypercube gives them resilience against certain types of
attack. We present new quantum algorithms that solve the hidden shift problems
for several well-known classes of bent functions in polynomial time and with a
constant number of queries, while the classical query complexity is shown to be
exponential. Our approach uses a technique that exploits the duality between
bent functions and their Fourier transforms.Comment: 15 pages, 1 figure, to appear in Proceedings of the 21st Annual
ACM-SIAM Symposium on Discrete Algorithms (SODA'10). This updated version of
the paper contains a new exponential separation between classical and quantum
query complexit
MaxSAT Resolution and Subcube Sums
We study the MaxRes rule in the context of certifying unsatisfiability. We
show that it can be exponentially more powerful than tree-like resolution, and
when augmented with weakening (the system MaxResW), p-simulates tree-like
resolution. In devising a lower bound technique specific to MaxRes (and not
merely inheriting lower bounds from Res), we define a new proof system called
the SubCubeSums proof system. This system, which p-simulates MaxResW, can be
viewed as a special case of the semialgebraic Sherali-Adams proof system. In
expressivity, it is the integral restriction of conical juntas studied in the
contexts of communication complexity and extension complexity. We show that it
is not simulated by Res. Using a proof technique qualitatively different from
the lower bounds that MaxResW inherits from Res, we show that Tseitin
contradictions on expander graphs are hard to refute in SubCubeSums. We also
establish a lower bound technique via lifting: for formulas requiring large
degree in SubCubeSums, their XOR-ification requires large size in SubCubeSums
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