I'm not sure what you mean by "without carry" - if you don't want one, don;t calculate one.
What datatype are the values in - if it's an int (which is a 32 bit type) then you'll have no problem with numbers being negative, and you can simply add them up in either an int or a long.
If the values are of type short (which is 16 bit) then the values could be negative. You'd need to check the most significant bit, and if it's set, treat the number especially before adding it to the sum.
I was talking about 16 bit numbers where the most significant bit (MSB) is set , i.e., negative numbers as far as the Java short type is concerned. Since their actual value is between 32768 and 65536 (and not between -32768 and -1 as Java would think) you'd need to handle them especially.
But since you're adding bytes, this can't happen. The only thing to watch out for is if the sum becomes larger than 32767 because a short type will then become negative. This may be impossible unless you have a lot of numbers you're adding up, though.