Barrel code on 513T

[MK111]

Guys, can you be specific about the locations of these marks? Viewable with the stock assembled or not?

When you hold the rifle with the muzzel pointing to you left that would be the left side of the gun. The left side of Remington rifles on the barrel at the action is the Mfg code date stampings.
Look at the first picture to understand what I am saying. The first letter shown "O" is the month in this case July. Remington used the letters BLACKPOWDERX as these 12 letters do not repeat themselves for the twelve months of the year. As I see it the "O" you are using is not the year but the month.
Try as hard as I can I can not make out what follows the "O" stamping. Try using a magnifier of some sort to figure out the next letters. Should be a "K" for 1941 "L" for 1942 "MM" for 1943 and "NN" for 1944. The complete chart will be found on the Remington collector site or in the back of a Blue Book on gun value.
Keeping in mind that maybe three letters maybe stamped and the last letter making no sense in the dating codes. That last letter is just some sort of proof stamping code.
On the right side of the barrel those are proof stamping or just other codes but not date coding stamps. I hope this is of helps to understanding Remington's date codes.