1,853 research outputs found
The weight enumerator polynomials of the lifted codes of the projective Solomon-Stiffler codes
Determining the weight distribution of a code is an old and fundamental topic
in coding theory that has been thoroughly studied. In 1977, Helleseth,
Kl{\o}ve, and Mykkeltveit presented a weight enumerator polynomial of the
lifted code over of a -ary linear code with
significant combinatorial properties, which can determine the support weight
distribution of this linear code. The Solomon-Stiffler codes are a family of
famous Griesmer codes, which were proposed by Solomon and Stiffler in 1965. In
this paper, we determine the weight enumerator polynomials of the lifted codes
of the projective Solomon-Stiffler codes using some combinatorial properties of
subspaces. As a result, we determine the support weight distributions of the
projective Solomon-Stiffler codes. In particular, we determine the weight
hierarchies of the projective Solomon-Stiffler codes.Comment: This manuscript was first submitted on September 9, 202
The Weight Hierarchies of Linear Codes from Simplicial Complexes
The study of the generalized Hamming weight of linear codes is a significant
research topic in coding theory as it conveys the structural information of the
codes and determines their performance in various applications. However,
determining the generalized Hamming weights of linear codes, especially the
weight hierarchy, is generally challenging. In this paper, we investigate the
generalized Hamming weights of a class of linear code \C over \bF_q, which
is constructed from defining sets. These defining sets are either special
simplicial complexes or their complements in \bF_q^m. We determine the
complete weight hierarchies of these codes by analyzing the maximum or minimum
intersection of certain simplicial complexes and all -dimensional subspaces
of \bF_q^m, where 1\leq r\leq {\rm dim}_{\bF_q}(\C)
Polynomial integrality gaps for strong SDP relaxations of Densest k
The Densest k-subgraph problem (i.e. find a size k subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for Densest k-subgraph: the current best algorithm gives an ≈ O(n 1/4) approximation, while even showing a small constant factor hardness requires significantly stronger assumptions than P ̸ = NP. In addition to interest in designing better algorithms, a number of recent results have exploited the conjectured hardness of Densest k-subgraph and its variants. Thus, understanding the approximability of Densest k-subgraph is an important challenge. In this work, we give evidence for the hardness of approximating Densest k-subgraph within polynomial factors. Specifically, we expose the limitations of strong semidefinite programs from SDP hierarchies in solving Densest k-subgraph. Our results include: • A lower bound of Ω ( n 1/4 / log 3 n) on the integrality gap for Ω(log n / log log n) rounds of the Sherali-Adams relaxation for Densest k-subgraph. This also holds for the relaxation obtained from Sherali-Adams with an added SDP constraint. Our gap instances are i
On Universal Prediction and Bayesian Confirmation
The Bayesian framework is a well-studied and successful framework for
inductive reasoning, which includes hypothesis testing and confirmation,
parameter estimation, sequence prediction, classification, and regression. But
standard statistical guidelines for choosing the model class and prior are not
always available or fail, in particular in complex situations. Solomonoff
completed the Bayesian framework by providing a rigorous, unique, formal, and
universal choice for the model class and the prior. We discuss in breadth how
and in which sense universal (non-i.i.d.) sequence prediction solves various
(philosophical) problems of traditional Bayesian sequence prediction. We show
that Solomonoff's model possesses many desirable properties: Strong total and
weak instantaneous bounds, and in contrast to most classical continuous prior
densities has no zero p(oste)rior problem, i.e. can confirm universal
hypotheses, is reparametrization and regrouping invariant, and avoids the
old-evidence and updating problem. It even performs well (actually better) in
non-computable environments.Comment: 24 page
2D growth processes: SLE and Loewner chains
This review provides an introduction to two dimensional growth processes.
Although it covers a variety processes such as diffusion limited aggregation,
it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner
evolutions (SLE) which are Markov processes describing interfaces in 2D
critical systems. It starts with an informal discussion, using numerical
simulations, of various examples of 2D growth processes and their connections
with statistical mechanics. SLE is then introduced and Schramm's argument
mapping conformally invariant interfaces to SLE is explained. A substantial
part of the review is devoted to reveal the deep connections between
statistical mechanics and processes, and more specifically to the present
context, between 2D critical systems and SLE. Some of the SLE remarkable
properties are explained, as well as the tools for computing with SLE. This
review has been written with the aim of filling the gap between the
mathematical and the physical literatures on the subject.Comment: A review on Stochastic Loewner evolutions for Physics Reports, 172
pages, low quality figures, better quality figures upon request to the
authors, comments welcom
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