General questions about memories, configuration and FPGA systems.

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Could I use CFI Flash for configuring automatically the FPGA and store different configuration codes and when needed start with different one ?

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AFAIK yes. Check how's the design made on Cyclone III development kit.

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Is it possible to store there also the code for NIOS that would be copied to SRAM or SDRAM? How copying is done?

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AFAIK yes. You create a small bootloader, which runs from on-chip ram (Your reset vector must point there). Then that bootloader copies data from flash to sram/sdram and jumps to that address.