On the existence of extremal Type II ℤ_{2𝕜}-codes

Abstract

For lengths 8, 16 and 24, it is known that there is an extremal Type II Z 2k -code for every positive integer k. In this paper, we show that there is an extremal Type II Z 2k -code of lengths 32, 40, 48, 56 and 64 for every positive integer k. For length 72, it is also shown that there is an extremal Type II Z 4k -code for every positive integer k with k ≥ 2.

On Automorphisms of Extremal Even Unimodular Lattices

“…Since the description there is slightly incorrect and no explicit construction of the automorphism groups is available in the literature, the construction of the automorphism groups is given in the next theorem. A third lattice, P 48n has been found by the author in [19], where it was proved that the normalizer of the subgroup SL 2 (13) in Aut(P 48n ) is (SL 2 (13)Y SL 2 (5)).2 2 . Using the classification of finite simple groups one may obtain the full automorphism group of these three lattices:…”

“…In [13] the authors construct two extremal even unimodular lattices A 14 (C 14,48 ) and A 46 (C 46,48 ) of dimension 48 as code lattices of self-dual codes of length 48. From the construction one also obtains a certain subgroup U of order 48 (the monomial automorphism group of the codes C 2p,48 ≤ Z/(2pZ) 48 , with monomial entries ±1) which can be read off from the construction of the codes with weighing matrices.…”

“…One gets γ n = 4δ 2/n n . The table below gives upper bounds b n for γ n : n b n n b n n b n n b n n b n 7 1.8115 13 The automorphism group Aut(L) := {σ ∈ O(R n , (, )) | σ(L) = L} respects the length and hence acts on Min(L). Taking matrices with respect to the lattice basis B, we obtain Aut(L) ≤ GL n (Z) is a finite integral matrix group.…”

An even extremal lattice of rank 64

“…The order of the automorphism group is at least 587520. In [3] and [4], the existence of extremal Type II Z 2k -codes of length 64 was shown. The isomorphism classes and the automorphism groups of the associated extremal lattices are, however, not clear.…”

Extremal Type I $\mathbb{Z}_k$-codes and $k$-frames of odd unimodular lattices

“…Type II Z 2k -codes were defined in [2] as a class of self-dual codes, which are related to even unimodular lattices. If C is a Type II Z 2k -code of length n ≤ 136, then we have the bound on the minimum Euclidean weight d E (C) of C as follows: d E (C) ≤ 4k n 24 + 4k for every positive integer k (see [21]).…”