105 research outputs found
The weight enumerators for certain subcodes of the second order binary Reed-Muller codes
In this paper we obtain formulas for the number of codewords of each weight in several classes of subcodes of the second order Reed-Muller codes. Our formulas are derived from the following results: (i) the weight enumerator of the second order RM code, as given by Berlekamp-Sloane (1970), (ii) the MacWilliams-Pless identities, (iii) a new result we present here (Theorem 1), (iv) the Carlitz-Uchiyama (1957) bound, and, (iv′) the BCH bound.The class of codes whose weight enumerators are determined includes subclasses whose weight enumerators were previously found by Kasami (1967–1969) and Berlekamp(1968a, b)
Electromagnetic superconductivity of vacuum induced by strong magnetic field: numerical evidence in lattice gauge theory
Using numerical simulations of quenched SU(2) gauge theory we demonstrate
that an external magnetic field leads to spontaneous generation of quark
condensates with quantum numbers of electrically charged rho mesons if the
strength of the magnetic field exceeds the critical value eBc = 0.927(77) GeV^2
or Bc =(1.56 \pm 0.13) 10^{16} Tesla. The condensation of the charged rho
mesons in strong magnetic field is a key feature of the magnetic-field-induced
electromagnetic superconductivity of the vacuum.Comment: 14 pages, 5 figures, 2 tables, elsarticle style; continuum limit is
analyzed, best fit parameters are presented in Table 2, published versio
Production of gliders by collisions in Rule 110
We investigate the construction of all the periodic structures or “gliders” up to now known in the evolution space of the one-dimensional cellular automaton Rule 110. The production of these periodic structures is developed and presented by means of glider collisions. We provide a methodology based on the phases of each glider to establish the necessary conditions for controlling and displaying the collisions of gliders from the initial configuration
Efficient implementation of a CCA2-secure variant of McEliece using generalized Srivastava codes
International audienceIn this paper we present efficient implementations of McEliece variants using quasi-dyadic codes. We provide secure parameters for a classical McEliece encryption scheme based on quasi-dyadic generalized Srivastava codes, and successively convert our scheme to a CCA2-secure protocol in the random oracle model applying the Fujisaki-Okamoto transform. In contrast with all other CCA2-secure code-based cryptosystems that work in the random oracle model, our conversion does not require a constant weight encoding function. We present results for both 128-bit and 80-bit security level, and for the latter we also feature an implementation for an embedded device
Improving the Berlekamp Algorithm for Binomials x n − a
In this paper, we describe an improvement of the Berlekamp algorithm, a method for factoring univariate polynomials over finite fields, for binomials xn −a over finite fields Fq. More precisely, we give a deterministic algorithm for solving the equation h(x)q≡h(x) (mod xn−a) directly without applying the sweeping-out method to the corresponding coefficient matrix. We show that the factorization of binomials using the proposed method is performed in O˜, (n log q) operations in Fq if we apply a probabilistic version of the Berlekamp algorithm after the first step in which we propose an improvement. Our method is asymptotically faster than known methods in certain areas of q, n and as fast as them in other areas
Growth and Decay in Life-Like Cellular Automata
We propose a four-way classification of two-dimensional semi-totalistic
cellular automata that is different than Wolfram's, based on two questions with
yes-or-no answers: do there exist patterns that eventually escape any finite
bounding box placed around them? And do there exist patterns that die out
completely? If both of these conditions are true, then a cellular automaton
rule is likely to support spaceships, small patterns that move and that form
the building blocks of many of the more complex patterns that are known for
Life. If one or both of these conditions is not true, then there may still be
phenomena of interest supported by the given cellular automaton rule, but we
will have to look harder for them. Although our classification is very crude,
we argue that it is more objective than Wolfram's (due to the greater ease of
determining a rigorous answer to these questions), more predictive (as we can
classify large groups of rules without observing them individually), and more
accurate in focusing attention on rules likely to support patterns with complex
behavior. We support these assertions by surveying a number of known cellular
automaton rules.Comment: 30 pages, 23 figure
Computing Naturally in the Billiard Ball Model
Fredkin's Billiard Ball Model (BBM) is considered one of the fundamental
models of collision-based computing, and it is essentially based on elastic
collisions of mobile billiard balls. Moreover, fixed mirrors or reflectors are
brought into the model to deflect balls to complete the computation. However,
the use of fixed mirrors is "physically unrealistic" and makes the BBM not
perfectly momentum conserving from a physical point of view, and it imposes an
external architecture onto the computing substrate which is not consistent with
the concept of "architectureless" in collision-based computing. In our initial
attempt to reduce mirrors in the BBM, we present a class of gates: the
m-counting gate, and show that certain circuits can be realized with few
mirrors using this gate. We envisage that our findings can be useful in future
research of collision-based computing in novel chemical and optical computing
substrates.Comment: 10 pages, 7 figure
On the generalized linear equivalence of functions over finite fields
In this paper we introduce the concept of generalized linear equivalence between functions defined over finite fields; this can be seen as an extension of the classical criterion of linear equivalence, and it is obtained by means of a particular geometric representation of the functions. After giving the basic definitions, we prove that the known equivalence relations can be seen as particular cases of the proposed generalized relationship and that there exist functions that are generally linearly equivalent but are not such in the classical theory. We also prove that the distributions of values in the Difference Distribution Table (DDT) and in the Linear Approximation Table (LAT) are invariants of the new transformation; this gives us the possibility to find some Almost Perfect Nonlinear (APN) functions that are not linearly equivalent (in the classical sense) to power functions, and to treat them accordingly to the new formulation of the equivalence criterion
Localization dynamics in a binary two-dimensional cellular automaton: the Diffusion Rule
We study a two-dimensional cellular automaton (CA), called Diffusion Rule
(DR), which exhibits diffusion-like dynamics of propagating patterns. In
computational experiments we discover a wide range of mobile and stationary
localizations (gliders, oscillators, glider guns, puffer trains, etc), analyze
spatio-temporal dynamics of collisions between localizations, and discuss
possible applications in unconventional computing.Comment: Accepted to Journal of Cellular Automat
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