This is a classical problem which goes under the name "non-coherent detection". Here are some facts:

1. Capacity of the block fading channel is not achieved using pilots.

2. Capacity of the channel is not achieved using Gaussian data.

3. Capacity of the channel is generally unknown.

4. Only capacity expansions in the low and high SNR regime are known to date.

5. With Gaussian data, the situation is analyzed in e.g. "forgot the title" by Rusek, Lozano, and Jindal" TIT 2012.

6. Input block should optimally consist of a Haar unitary matrix plus a certain, to date unknown, pdf of the singular values.

7. Grassmanian codebooks are usually used in practical situations.

8. With pilots: estimating the channel and using the estimate as it is correct is analyzed in e.g. "forgot the title" by Lapidoth and Shamai, TIT 2002

This is a classical problem which goes under the name "non-coherent detection". Here are some facts:

1. Capacity of the block fading channel is not achieved using pilots.

2. Capacity of the channel is not achieved using Gaussian data.

3. Capacity of the channel is generally unknown.

4. Only capacity expansions in the low and high SNR regime are known to date.

5. With Gaussian data, the situation is analyzed in e.g. "forgot the title" by Rusek, Lozano, and Jindal" TIT 2012.

6. Input block should optimally consist of a Haar unitary matrix plus a certain, to date unknown, pdf of the singular values.

7. Grassmanian codebooks are usually used in practical situations.

8. With pilots: estimating the channel and using the estimate as it is correct is analyzed in e.g. "forgot the title" by Lapidoth and Shamai, TIT 2002