Masaaki Kitazume

Extremal self-dual codes of length 64 through neighbors and covering radii

We construct extremal singly even self-dual [64,32,12] codes with weight enumerators which were not known to be attainable. In particular, we find some codes whose shadows have minimum weight 12. By considering their doubly even neighbors, extremal doubly even self-dual [64,32,12] codes with covering radius 12 are constructed for the first time.

Ternary Code Construction of Unimodular Lattices and Self-Dual Codes over \Bbb Z 6

We revisit the construction method of even unimodular lattices using ternary self-dual codes given by the third author (M. Ozeki, in Théorie des nombres, J.-M. De Koninck and C. Levesque (Eds.) (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 772–784), in order to apply the method to odd unimodular lattices and give some extremal (even and odd) unimodular lattices explicitly. In passing we correct an error on the condition for the minimum norm of the lattices of dimension a multiple of 12. As the results of our present research, extremal odd unimodular lattices in dimensions 44, 60 and 68 are constructed for the first time. It is shown that the unimodular lattices obtained by the method can be constructed from some self-dual ℤ6-codes. Then extremal self-dual ℤ6-codes of lengths 44, 48, 56, 60, 64 and 68 are constructed.

On some self-dual codes and unimodular lattices in dimension 48

In this paper, binary extremal self-dual codes of length 48 and extremal unimodular lattices in dimension 48 are studied through their shadows and neighbors. We relate an extremal singly even self-dual [48, 24, 10] code whose shadow has minimum weight 4 to an extremal doubly even self-dual [48, 24, 12] code. It is also shown that an extremal odd unimodular lattice in dimension 48 whose shadow has minimum norm 2 relates to an extremal even unimodular lattice. Extremal singly even self-dual [48, 24, 10] codes with shadows of minimum weight 8 and extremal odd unimodular lattice in dimension 48 with shadows of minimum norm 4 are investigated.

The radical subgroups of the Fischer simple groups

A nontrivial p-subgroupR of a finite groupG is called radical if R = Op(NG(R)). The present paper is the second one of the series of papers [8, 16–18], which project to classify the radical p-subgroups of all sporadic simple groups and every prime p, except possibly the Monster and the baby Monster for p = 2. The importance of such classifications is twofold. One is related with the Dade conjecture: the classification is the first step to find suitable classes of radical chains to which the conjecture are verified (see [11,12] for recent progress, in which the results of [16–18] are used). The second and more natural one to the second author is its connection with p-local geometries (see [15, Section 1] for the detail): the classification is the first step to examine the simplicial complex ∆(Bp(G)) of chains of radical p-subgroups, which would be a crucial object to understand “ p-local geometries” for finite simple (specifically sporadic) groups in a unified point of view. The radical groups of sporadic simple groups of characteristic-2 type are classified in [16]. In this paper, we classify the radical 2and 3-subgroups of

Niemeier lattices and Type II codes over Z 4

Abstract The 23 Niemeier lattices are constructed by Construction A 4 . Explicit generator matrices and symmetrized weight enumerators for the relevant codes are given.

On a 5-design related to an extremal doubly even self-dual code of length 72

It is shown that if there is a self-orthogonal 5-(72,16,78) design, then the rows of its block-point incidence matrix generate an extremal doubly even self-dual code of length 72. In other words, a putative extremal doubly even self-dual code of length 72 is generated by the codewords of minimum weight.

Z4-Code Constructions for the Niemeier Lattices and their Embeddings in the Leech Lattice

In this note, we give Z4-code constructions of the Niemeier lattices, showing their embedding in the Leech lattice. These yield an alternative proof of a recent result by Dong et al..

Z6-Code Constructions of the Leech Lattice and the Niemeier Lattices

In this paper, we construct many new extremal Type II Z6-codes of length 24, and consequently we show that there is at least one extremal Type II Z6-code C of length 24 such that the binary and ternary reductions of C are B and T, respectively, for every binary Type II code B and every extremal ternary self-dual code T. These codes give more Z6-code constructions of the Leech lattice. It is also shown that every Niemeier lattice contains a (4 k2+ 2k+ 6)-frame for every integer k.

Moonshine vertex operator algebra as L(12,0)⊗L(710,0)⊗L(45,0)⊗L(1,0)-modules

Abstract In this article, we study the structure of the famous Moonshine vertex operator algebra V ♮ as a module for a certain vertex operator subalgebra isomorphic to (L( 1 2 ,0)⊗L( 7 10 ,0)⊗L( 4 5 ,0)⊗L(1,0)) ⊗8 . As our main result, we obtain a complete decomposition of V ♮ associated with this algebra. Our method is based on an embedding of the lattice ( 2 A 3 ) 8 into the Leech lattice. Actually, we construct certain vertex operator algebras using some Z 8 and Z 2 × Z 2 codes. A decomposition of the Moonshine vertex operator algebra is obtained by a careful study of their representations.

Self-Dual Codes Related to the Rudvalis Group

In this note, we will show that there exist exactly three self-dual codes with the automorphism group isomorphic to the Rudvalis group up to equivalence. Furthermore we will give some descriptions of codewords which span these codes. Especially, we will describe some codewords by using a relation between the rank 3 graph of the Rudvalis group and the Hoffman–Singleton graph.