There are Exactly Two Non-Equivalent \([20,5,12;3]\)-Codes

Hill and Newton showed that there exists a \([20, 6, 12; 3]\)-code, and that the weight distribution of a \([20,5, 12; 3]\)-code is unique. However, it is unknown whether or not a code with these parameters is unique. Recently, Hamada and Helleseth showed that a \([19, 4, 12; 3]\)-code is unique up to equivalence, and characterized this code using a characterization of \(\{21, 6; 3, 3\}\)-minihypers. The purpose of this paper is to show, using the geometrical structure of the \([19, 4, 12; 3]\)-code, that exactly two non-isomorphic \([20, 5, 12; 3]\)-codes exist.