arXiv:1111.2637v1 [math.CO] 11 Nov 2011 Some Extremal Self-Dual Codes and Unimodular Lattices in Dimension 40 ∗ Masaaki Harada † November 14, 2011 Abstract In this paper, binary extremal singly even self-dual codes of length 40 and extremal odd unimodular lattices in dimension 40 are studied. We give a classification of extremal odd unimodular lattices in dimen- sion 40 with shadows having 80 vectors of norm 2 and extremal singly even self-dual codes of length 40 with shadows of minimum weight 4 through their relationships with doubly even self-dual codes of length 40. We also discuss related extremal self-dual Z 4 -codes and extremal even self-dual additive F 4 -codes. 1 Introduction Self-dual codes and unimodular lattices are studied from several viewpoints (see [15] for an extensive bibliography). Also many relationships between self- dual codes and unimodular lattices are known and there are similar situations between two subjects. In this paper, binary singly even self-dual codes of length 40 and odd unimodular lattices in dimension 40 are studied using relationships with binary doubly even self-dual codes of length 40. This study gives new examples of similar situations between self-dual codes and unimodular lattices. * This work was supported by JST PRESTO program. † Department of Mathematical Sciences, Yamagata University, Yamagata 990–8560, Japan, and PRESTO, Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332–0012, Japan. email: mharada@sci.kj.yamagata-u.ac.jp 1