Extremal self-dual \([40,20,8]\) codes with covering radius 7 (Q1827585)

\textit{E. Assmus} and \textit{V. Pless} [IEEE Trans. Inf. Theory 29, 359--363 (1983; Zbl 0504.94028)] showed that the minimum possible covering radius of an extremal doubly-even binary \([40,20,8]\) code is 6, 7, or 8; \textit{M. Harada} and \textit{M. Ozeki} [Discrete Math. 215, 271--281 (2000; Zbl 0974.94027)] found such a code with covering radius 7. Nonexistence of codes with covering radius 6 has been mentioned in the literature, but these results were withdrawn by \textit{M. Ozeki} [IEEE Trans. Inf. Theory 48, 547--556 (2002; Zbl 1064.94020 In the current paper an extremal doubly-even binary \([40,20,8]\) code with covering radius 7 is presented which is not equivalent to the code presented by Harada and Ozeki (loc. cit.). Singly-even codes with these parameters are also discussed, as well as some related 1-designs.