simple socket server connection established, but cannot ping/telnet

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How can I verify Signal Integrity and FPGA timings?

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Time Quest for FPGA timings. If you don't know how to use it, you'll have to learn it.

Signal integrity depends on how well is designed your pcb. Something can be evaluated with a good oscilloscope but first of all you must check if everything complies with component specifications (i.e. capacitive decoupling, ground/supply layers, board traces routing, ...)

In your situation I'll especially focus on PHY signals, both on fpga and wire sides.