There certainly are a number of options available for growing peppers. After doing A LOT of reading on the pot forums and a some trial and error a few years back I went with coco coir as the grow medium. I know it's not the kind of hydroponics you were asking about since it could be considered more 'hydro-like' than anything else.
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I use a variation of the General Hydroponics recipe that was formulated by a poster by the name of GreatfulH3ad. It is DTW and is mixed as:
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Nutrient mixture of:
 - 1.5ml GH Armor Si (I substituted this in for pH Down)
 - 6ml GH Micro
 - 9ml GH Bloom
 - 4l RO water
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This nutrient mixture gives me a ph of 5.7 and 700 ppm. The NPK falls in at 100-100-200 ppm. I use it throughout the grow. Personally, I mix the nutrients as I need them and water by hand so I don't need to worry about ph swing or temperature variation, but I could see the benefits of a nicely automated drip system if you have space to set it up.Â
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Container and grow medium:
 - 5g fiber root pouch
 - Sifted coco coir to remove the fine pith
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I have grown in plastic containers in the past, but I've found the that the fiber pots are better in that they allow for a healthy root mass that air prunes itself rather than endless larger roots that wrap around the pot. They also seem to resist the heat better than a black plastic pot and the root zone doesn't get as hot in the sun.
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In terms of saving space, I've found that I don't have to grow as many plants in order to get a decent harvest when using this method simply because the pepper production is much higher than I've had with other methods in the past. I'm sure that can be said for most, if not all hydro methods.
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I have a few old grow logs here from back in 2016 should you want to check them out.