Is it possible to have negative corrorion rate and weight loss, meaning specimen [steel] gained weight after immersion in electroyte solution?
In the data cited in your reference, corrosion is measured by weight loss or gain. In the case of the steel coupon immersed in sodium carbonate, the weight gain is not due to "negative corrosion", but likely due to a tightly-adhering film of oxides that was not removed in cleaning. The oxides are made from the iron of the coupon plus oxygen from water, which is greater than the mass of the iron alone. (Normally, cleaning the coupon removes loosely adhering oxides, so there is a weight loss.)

According to Toyin, "the corrosion inhibition efficiency of $\ce{Na2CO3}$... indicates that a protective film is formed on the metal surface. The passivation of the mild steel may be caused by the formation of $\ce{Fe2O3}$  and $\ce{Fe3O4}$."

That said, it does appear that high carbon concentrations may be effective in greatly reducing the corrosion of mild steel.