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

Abstract

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.