Mod 2 Hopf ring for the sphere spectrum

An entry in row \( k \) and column \( n \) of the table below is an element of \( H_n( \underline{\mathrm{S}}_k ; \mathbb{F}_2) \), the mod 2 Hopf ring for the sphere spectrum. Only some \(*\)-indecomposable elements are shown. Juxtaposition and exponentiation denote circle products (\( xy = x \circ y\) and \(x^n = x \circ x \circ \cdots x\) with \( n \) factors). Compact notation: \( E(a,b) \) is an abbreviation for \( E_a \circ E_b = Q^a[1] \circ Q^b[1] \).