  Recent work has established that when transmitter Alice wishes to communicate
reliably to recipient Bob without detection by warden Willie, with additive
white Gaussian noise (AWGN) channels between all parties, communication is
limited to $\mathcal{O}(\sqrt{n})$ bits in $n$ channel uses. However, this
assumes Willie has an accurate statistical characterization of the channel.
When Willie has uncertainty about such and his receiver is limited to a
threshold test on the received power, Alice can transmit covertly with a power
that does not decrease with $n$, thus conveying $\mathcal{O}(n)$ bits covertly
and reliably in $n$ uses of an AWGN channel. Here, we consider covert
communication of $\mathcal{O}(n)$ bits in $n$ channel uses while generalizing
the environment and removing any restrictions on Willie's receiver. We assume
an uninformed "jammer" is present to help Alice, and we consider AWGN and block
fading channels. In some scenarios, Willie's optimal detector is a threshold
test on the received power. When the channel between the jammer and Willie has
multiple fading blocks per codeword, a threshold test on the received power is
not optimal. However, we establish that Alice can remain covert with a transmit
power that does not decrease with $n$ even when Willie employs an optimal
detector.
