The fourth Singer algebraic transfer is a monomorphism at degrees d_{s,t} = 2^{s+t}+2^s-3 and n_{s,t} = 2^{s+t}+2^s-2, assuming the paper's explicit but unverified dimension computations are correct.
Boardman,Modular representations on the homology of power of real projective space, in: M
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An application of the hit problem to the algebraic transfer
The fourth Singer algebraic transfer is a monomorphism at degrees d_{s,t} = 2^{s+t}+2^s-3 and n_{s,t} = 2^{s+t}+2^s-2, assuming the paper's explicit but unverified dimension computations are correct.