Code Loops in Dimension at Most 8

Publication Date

11-24-2016

Document Type

Article

Organizational Units

College of Natural Science and Mathematics, Mathematics

Keywords

Code loop, Doubly even code, Trilinear alternating form, Moufang loop, General linear group, Sporadic group, The Monster group

Abstract

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 classifications of trilinear alternating forms over the field of 2 elements, we enumerate code loops of dimension d=dim⁡(V)≤8" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">d=dim⁡(V)≤8 (equivalently, of order 2d+1≤512" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">2d+1≤512) up to isomorphism. There are 767 code loops of order 128, and 80826 of order 256, and 937791557 of order 512.

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