I remember hearing somewhere, it may have been in the roman military, that if a soldier got shot with an arrow and was in his back, it was deemed dishonorable because he was running away and there was some sort of punishment. Can anyone clarify or maybe further explain this?
The Shinsengumi (a samurai police unit which existed for a short time under the authority of the last shogun of Japan) had the rule for themselves that a member who was injured in the back during battle would be assumed to have been fleeing in cowardice, and would be therefore forced to commit seppuku (ritual suicide.) They are the only group I've ever heard of who had such a rule, although there may have been others.
You seem to understand. Not sure what you want clarified.
If they had an arrow in their back, that means they were running away from combat. Dying a dishonourable death by being a coward. Cowardice is dishonourable.