Code loops in dimension at most 8

@article{OBrien2017CodeLI,
  title={Code loops in dimension at most 8},
  author={Eamonn A. O'Brien and Petr Vojtvechovsk'y},
  journal={arXiv: Group Theory},
  year={2017}
}
Code loops are certain Moufang $2$-loops constructed from doubly even binary codes that play an important role in the construction of local subgroups of sporadic groups. More precisely, code loops are central extensions of the group of order $2$ by an elementary abelian $2$-group $V$ in the variety of loops such that their squaring map, commutator map and associator map are related by combinatorial polarization and the associator map is a trilinear alternating form. Using existing… Expand
5 Citations

Tables from this paper

Weights of the $\mathbb{F}_{q}$-forms of $2$-step splitting trivectors of rank $8$ over a finite field
Grassmann codes are linear codes associated with the Grassmann variety $G(\ell,m)$ of $\ell$-dimensional subspaces of an $m$ dimensional vector space $\mathbb{F}_{q}^{m}.$ They were studied by NoginExpand
(Re)constructing Code Loops
TLDR
This expository article aims to highlight an experimental approach to computing in code loops, by a combination of a small amount of precomputed information and making use of the rich identities that code loops’ twisted cocycles satisfy. Expand
Classification of $k$-forms on ${\bf R}^n$ and the existence of associated geometry on manifolds
In this paper we survey methods and results of classification of $k$-forms (resp. $k$-vectors on ${\bf R}^n$), understood as description of the orbit space of the standard ${\bf GL}(n, {\bfExpand
Classification of 9-dimensional trilinear alternating forms over GF(2)
TLDR
Two new invariants are introduced that are sufficient to distinguish between all trilinear alternating forms in dimension 9 over GF(2) and to prove the completeness of the list of forms. Expand
Variety of loops generated by code loops
In this work, we construct free infinitely generated Moufang loop in the variety generated by code loops and find the minimal set of identities that define this variety. We apply this construction toExpand

References

SHOWING 1-10 OF 35 REFERENCES
Moufang Loops that Share Associator and Three Quarters of Their Multiplication Tables
Two constructions due to Dr\'apal produce a group by modifying exactly one quarter of the Cayley table of another group. We present these constructions in a compact way, and generalize them toExpand
Explicit constructions of code loops as centrally twisted products
  • T. Hsu
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
  • 2000
Code loops are certain Moufang loop extensions of doubly even binary codes which are useful in finite group theory (e.g. Conway's construction of the Monster). We give several methods for explicitlyExpand
Local subgroups of the Monster and odd code loops
The main result of this work is an explicit construction of p-local subgroups of the Monster, the largest sporadic simple group. The groups constructed are the normalizers in the Monster of certainExpand
Moufang loops of class 2 and cubic forms
  • T. Hsu
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
  • 2000
We classify finite Moufang loops which are centrally nilpotent of class 2 in terms of certain cubic forms, concentrating on small Frattini Moufang loops, or SFMLs, which are Moufang loops L with aExpand
Direct construction of code loops
  • G. Nagy
  • Mathematics, Computer Science
  • Discret. Math.
  • 2008
TLDR
This paper presents a global construction of the loop, where the correspondence between the concepts of Moufang loops and groups with triality is applied. Expand
Moufang loops of small order. I
The main result of this paper is the determination of all nonassociative Moufang loops of orders *31. Combinatorial type methods are used to consider a number of cases which lead to the discovery ofExpand
Combinatorial aspects of code loops
The existence and uniqueness (up to equivalence) of code loops was first established by R. Griess. Nevertheless, the explicit construction of code loops remained open until T. Hsu introduced theExpand
Classification of 8-Dimensional Trilinear Alternating Forms over GF(2)
Let V be an n-dimensional vector space over a finite field and let f be a trilinear alternating form over V. For such forms we introduce a new invariant called radical polynomial and investigate itsExpand
The Moufang loops of order 64 and 81
TLDR
This work classifies Moufang loops of order 64 and 81 up to isomorphism, using a linear algebraic approach to central loop extensions to identify nonassociative and commutative loops. Expand
Code loops in both parities
We present equivalent definitions of code loops in any characteristic p≠0. The most natural definition is via combinatorial polarization, but we also show how to realize code loops by linear codesExpand
...
1
2
3
4
...