188,963 research outputs found

    Trellis decoding complexity of linear block codes

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    In this partially tutorial paper, we examine minimal trellis representations of linear block codes and analyze several measures of trellis complexity: maximum state and edge dimensions, total span length, and total vertices, edges and mergers. We obtain bounds on these complexities as extensions of well-known dimension/length profile (DLP) bounds. Codes meeting these bounds minimize all the complexity measures simultaneously; conversely, a code attaining the bound for total span length, vertices, or edges, must likewise attain it for all the others. We define a notion of “uniform” optimality that embraces different domains of optimization, such as different permutations of a code or different codes with the same parameters, and we give examples of uniformly optimal codes and permutations. We also give some conditions that identify certain cases when no code or permutation can meet the bounds. In addition to DLP-based bounds, we derive new inequalities relating one complexity measure to another, which can be used in conjunction with known bounds on one measure to imply bounds on the others. As an application, we infer new bounds on maximum state and edge complexity and on total vertices and edges from bounds on span lengths

    The weight hierarchies and chain condition of a class of codes from varieties over finite fields

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    The generalized Hamming weights of linear codes were first introduced by Wei. These are fundamental parameters related to the minimal overlap structures of the subcodes and very useful in several fields. It was found that the chain condition of a linear code is convenient in studying the generalized Hamming weights of the product codes. In this paper we consider a class of codes defined over some varieties in projective spaces over finite fields, whose generalized Hamming weights can be determined by studying the orbits of subspaces of the projective spaces under the actions of classical groups over finite fields, i.e., the symplectic groups, the unitary groups and orthogonal groups. We give the weight hierarchies and generalized weight spectra of the codes from Hermitian varieties and prove that the codes satisfy the chain condition

    The SMEFTsim package, theory and tools

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    We report codes for the Standard Model Effective Field Theory (SMEFT) in FeynRules -- the SMEFTsim package. The codes enable theoretical predictions for dimension six operator corrections to the Standard Model using numerical tools, where predictions can be made based on either the electroweak input parameter set {α^ew,m^Z,G^F}\{\hat{\alpha}_{ew}, \hat{m}_Z, \hat{G}_F \} or {m^W,m^Z,G^F}\{\hat{m}_{W}, \hat{m}_Z, \hat{G}_F\}. All of the baryon and lepton number conserving operators present in the SMEFT dimension six Lagrangian, defined in the Warsaw basis, are included. A flavour symmetric U(3)5{\rm U}(3)^5 version with possible non-SM CP\rm CP violating phases, a (linear) minimal flavour violating version neglecting such phases, and the fully general flavour case are each implemented. The SMEFTsim package allows global constraints to be determined on the full Wilson coefficient space of the SMEFT. As the number of parameters present is large, it is important to develop global analyses on reduced sets of parameters minimizing any UV assumptions and relying on IR kinematics of scattering events and symmetries. We simultaneously develop the theoretical framework of a "W-Higgs-Z pole parameter" physics program that can be pursued at the LHC using this approach and the SMEFTsim package. We illustrate this methodology with several numerical examples interfacing SMEFTsim with MadGraph5. The SMEFTsim package can be downloaded at https://feynrules.irmp.ucl.ac.be/wiki/SMEFTComment: Corrected numerics of section 10.5.1, references added, minor changes and corrected typos. Version published in JHE

    Transients from Initial Conditions: A Perturbative Analysis

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    The standard procedure to generate initial conditions (IC) in numerical simulations is to use the Zel'dovich approximation (ZA). Although the ZA correctly reproduces the linear growing modes of density and velocity perturbations, non-linear growth is inaccurately represented because of the ZA failure to conserve momentum. This implies that it takes time for the actual dynamics to establish the correct statistical properties of density and velocity fields. We extend perturbation theory (PT) to include transients as non-linear excitations of decaying modes caused by the IC. We focus on higher-order statistics of the density contrast and velocity divergence, characterized by the S_p and T_p parameters. We find that the time-scale of transients is determined, at a given order p, by the spectral index n. The skewness factor S_3 (T_3) attains 10% accuracy only after a=6 (a=15) for n=0, whereas higher (lower) n demands more (less) expansion away from the IC. These requirements become much more stringent as p increases. An Omega=0.3 model requires a factor of two larger expansion than an Omega=1 model to reduce transients by the same amount. The predicted transients in S_p are in good agreement with numerical simulations. More accurate IC can be achieved by using 2nd order Lagrangian PT (2LPT), which reproduces growing modes up to 2nd order and thus eliminates transients in the skewness. We show that for p>3 this reduces the required expansion by more than an order of magnitude compared to the ZA. Setting up 2LPT IC only requires minimal, inexpensive changes to ZA codes. We suggest simple steps for its implementation.Comment: 37 pages, 10 figure
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