Z code sequencing


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I'm a coder for a diagnostic lab facility that specializes in cardiovascular and endocrine disease prevention. On our test requisition forms physicians are required to check off codes we have listed as indicators, or they may write in codes that are not pre listed on the form. I am only allowed to use 4 codes max, which means I spend some time prioritizing what goes into the system and what doesn't based on what tests were done and what codes are listed. On our request forms there is a personal and family history section that can be checked off ie. family hx of cardiovascular disease (Z82.49) and family hx of diabetes(Z83.3).
My question is this: ICD10 coding guidelines state that Z codes are to be listed last. Does this mean that if I have more than four codes listed, and say Z8249 is one of them, and it comes down to that or say a common cold, do I have to pick the cold as the fourth code?