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The aim of this work is to publicise some computational results involvingtables which contain $\mathcal{A}$-annihilated monomials, excluding squareclasses, in the Dyer-Lashof algebra and $H_*QS^0$; our computations go up todimension $1.1\times 10^7$ but the tables in this paper only announce resultsup to dimension $2^{17}=131072$ and full tables would be available uponrequest. The theoretical background for our computations is provided by work ofCurtis \cite{Curtis} and Wellington \cite{Wellington} on the$\mathcal{A}$-module structure of the Dyer-Lashof algebra as well as $H_*QS^0$.It seems to us that there is a workable algorithm to do these computationswhich we plan to announce in a future work, partly to avoid making this paperlonger than it is. We hope to receive feedback from the experts on thesecomputations and make our algorithm available as soon as we can. We hope thatthese tables provide a source for researchers in the field, as well as a poolof data to analyse the behaviour of these sequences, their distributions andother asymptotic behaviours. The problem of computing spherical classes in$H_*QS^0$ as well as the symmetric and non-symmetric hit problems have been ourmain motivations to pursue this project.

An algorithmic search for $\mathcal{A}$-annihilated classes in the  Dyer-Lashof algebra and $H_*QS^0$ I. Closed form for low lengths and tables  in low dimensions