Your strategy is correct as you will have to insert a linker between every epitope that you have designed for the vaccine. Also, are all the 20 epitopes required? You can refer the paper 'Optimization of codon pair use within the (GGGGS)3 linker sequence results in enhanced protein expression' which will give you a good insight on the linker sequence that has to be used. For codon optimisation, there are several softwares available like DNA works that is very helpful.

Thank you Mallika for your contribution. I will consult the article and software. Regarding the number of epitopes, we are just trying to have a good representative from common isolates to make a mutivalent vaccine

Thank you Andrei, appreciated

GS linker is not necessary to insert between T-cell epitopes.   GS linker is suitable to insert between B-cell epitopes and not between each pair.  You can use our approach as the successful example for such a design.  https://www.lens.org/lens/patent/WO_2011_110953_A2 

I'm not sure what you think you are buying by this strategy of targeting a large number of linear epitopes. Even with just a few, that's a variant of the older component peptide vaccine method that doesn't work very well. Perhaps you are trying to target HIV epitopes? Is that why you want to do this?

If you link together that many linear peptide sequences, what you will have, if you get expression, is an entirely new protein. How it would fold is a good question. You could try Rosetta or Robetta on it. 

http://robetta.bakerlab.org/  - High quality structure prediction. http://www.bioinfo.rpi.edu/bystrc/hmmstr/server.php

How that protein would relate to generating antibodies you want would be impossible to predict. There is no reason to think that the proteasome  would chop up that novel sequence into parts resembling your desired epitope. It might, but I would highly doubt that most of them would get presented as you hope by MHC.

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1783040/ - In figure 1, the little barrel shaped thing labeled 1, antigen processing, is the proteasome. Understanding it is key, because the way the proteasome breaks things up defines what can be presented to the immune system.  Note that the best epitopes are non-linear as a rule.

I'd expect that something out of that 20-linked protein would produce an antibody that sticks to some parts of the target protein(s) you are after. However, you could also produce some that are unique, and the affinity may not be all that great. I wouldn't want to  try that on humans, mainly because there is the potential risk of generating a rogue antibody that produces auto-immunity. Pretty low risk, but not out of the question. 

In any case, primates don't produce B-cell response well from DNA vaccination. It is sometimes termed the primate barrier. Most of it would be T-cell response. B-cell response will happen, but it won't be a strong response.

If this is for an animal vaccine and you want it to work, I would suggest simplicity. Express key proteins completely. Let the immune system decide what the best epitope is. But not knowing your goals, whether it is educational, etc. it's hard to say. One can sometimes learn more from doing strange and possibly useless things than doing a conservative, more likely to be viable project.