Codes which are ideals in abelian group algebras.

Take G to be any multiplicative group. Let [G] = and choose q to be a prime such that n and q are relatively prime. Let K denote the field of order q (i.e. GF(q)-K). We form the group algebra KG defined to be the set of all formal sums {mathematical formula} with multiplication and addition defined by {series of formulas}. A straightforward application of these definitions yields that KG is an associative algebra with multiplicative identity. In fact, the identity in the group G acts as the multiplicative identity in KG. Definition 1.1.1. A ring is said to satisfy the minimum chain condition if it satisfies-the following two properties: {more formulas} The dimension of KG over Kasa vector space is n, and every ideal of KG is a vector subspace. ·Therefore, KG satisfies the minimum chain condition. {formula} s. the set of all products of k elements in I). The radical of the ring, (denoted Rad(R)), is the sum of all nilpotent left ideals

How to estimate Fisher information matrices from simulations

The Fisher information matrix is a quantity of fundamental importance for information geometry and asymptotic statistics. In practice, it is widely used to quickly estimate the expected information available in a data set and guide experimental design choices. In many modern applications, it is intractable to analytically compute the Fisher information and Monte Carlo methods are used instead. The standard Monte Carlo method produces estimates of the Fisher information that can be biased when the Monte-Carlo noise is non-negligible. Most problematic is noise in the derivatives as this leads to an overestimation of the available constraining power, given by the inverse Fisher information. In this work we find another simple estimate that is oppositely biased and produces an underestimate of the constraining power. This estimator can either be used to give approximate bounds on the parameter constraints or can be combined with the standard estimator to give improved, approximately unbiased estimates. Both the alternative and the combined estimators are asymptotically unbiased so can be also used as a convergence check of the standard approach. We discuss potential limitations of these estimators and provide methods to assess their reliability. These methods accelerate the convergence of Fisher forecasts, as unbiased estimates can be achieved with fewer Monte Carlo samples, and so can be used to reduce the simulated data set size by several orders of magnitude.Comment: Supporting code available at https://github.com/wcoulton/CompressedFishe

Primordial information content of Rayleigh anisotropies

Anisotropies in the cosmic microwave background (CMB) are primarily generated by Thomson scattering of photons by free electrons. Around the time of recombination, the Thomson scattering probability quickly diminishes as the free electrons combine with protons to form neutral hydrogen. This production of neutral hydrogen enables a new type of scattering to occur: Rayleigh scattering of photons by hydrogen atoms. Unlike Thomson scattering, Rayleigh scattering is frequency dependent resulting in the generation of anisotropies with a different spectral dependence. Unfortunately the Rayleigh scattering efficiency rapidly decreases with the expansion of the neutral universe, with the result that only a small percentage of photons are scattered off of neutral hydrogen after recombination. Although the effect is very small, future CMB missions with higher sensitivity and improved frequency coverage are poised to measure the effect of Rayleigh scattering. The uncorrelated component of the Rayleigh anisotropies contains unique information on the primordial perturbations that could potentially be leveraged to expand our knowledge of the early universe. In this paper we explore whether measurements of Rayleigh scattering anisotropies can be used to constrain primordial non-Gaussianity and examine the hints of anomalies found by WMAP and Planck satellites. We show that the additional Rayleigh information has the potential to improve primordial non-Gaussianity constraints over pure Thompson constraints by 30%, or more. Primordial bispectra that are not of the local type benefit the most from these additional scatterings, which we attribute to the different scale dependence of the Rayleigh anisotropies. Unfortunately this different scaling means that Rayleigh measurements cannot be used to constrain anomalies or features on large scales. On the other hand, anomalies that may persist to smaller scales, such as the potential power asymmetry seen in WMAP and Planck, could be improved by the addition of Rayleigh scattering measurements.</p

Primordial information content of Rayleigh anisotropies

Anisotropies in the cosmic microwave background (CMB) are primarily generated by Thomson scattering of photons by free electrons. Around recombination, the Thomson scattering probability quickly diminishes as the free electrons combine with protons to form neutral hydrogen off which CMB photons can scatter through Rayleigh scattering. Unlike Thomson scattering, Rayleigh scattering is frequency dependent resulting in the generation of anisotropies with a different spectral dependence. Unfortunately the Rayleigh scattering efficiency rapidly decreases with the expansion of the neutral universe, with the result that only a small percentage of photons are scattered by neutral hydrogen. Although the effect is very small, future CMB missions with higher sensitivity and improved frequency coverage are poised to measure Rayleigh scattering signal. The uncorrelated component of the Rayleigh anisotropies contains unique information on the primordial perturbations that could potentially be leveraged to expand our knowledge of the early universe. In this paper we explore whether measurements of Rayleigh scattering anisotropies can be used to constrain primordial non-Gaussianity (NG) and examine the hints of anomalies found by WMAP and \textit{Planck} satellites. We show that the additional Rayleigh information has the potential to improve primordial NG constraints by $30\%$, or more. Primordial bispectra that are not of the local type benefit the most from these additional scatterings, which we attribute to the different scale dependence of the Rayleigh anisotropies. Unfortunately this different scaling means that Rayleigh measurements can not be used to constrain anomalies or features on large scales. On the other hand, anomalies that may persist to smaller scales, such as the potential power asymmetry seen in WMAP and \textit{Planck}, could be improved by the addition of Rayleigh measurements.Comment: 12 pages, 5 figure