Codes in Johnson graphs associated with quadratic forms over F2

Mark Ioppolo

doi:10.26182/5efd33ee8efef

Codes in Johnson graphs associated with quadratic forms over F₂

Mark Ioppolo

School of Physics, Mathematics and Computing

Research output: Thesis › Doctoral Thesis

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Abstract

This thesis studies strongly incidence-transitive codes in Johnson graphs associated with the 2-transitive actions of Sp(2n,2) of
degrees 2^2n-1±2^n-1. We construct two new infinite families of strongly incidence-transitive codes and demonstrate that they are
the only examples with codeword stabilisers contained in a geometric Aschbacher class. We construct two additional examples
using the fully deleted permutation modules for the symmetric group S_m with m=10 and show no further examples arise for
other values of m. If a codeword stabiliser is almost-simple then we show in most cases that the corresponding code cannot be
strongly incidence-transitive, though several possibilities remain open.

Original language	English
Qualification	Doctor of Philosophy
Awarding Institution	The University of Western Australia
Supervisors/Advisors	Bamberg, John, Supervisor Devillers, Alice, Supervisor Praeger, Cheryl, Supervisor
Thesis sponsors	Australian Government Research Training Program
Award date	10 Mar 2020
DOIs	https://doi.org/10.26182/5efd33ee8efef
Publication status	Unpublished - 2019

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THESIS - DOCTOR OF PHILOSOPHY - IOPPOLO, Mark Adam - 2019
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Cite this

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title = "Codes in Johnson graphs associated with quadratic forms over F2",

abstract = "This thesis studies strongly incidence-transitive codes in Johnson graphs associated with the 2-transitive actions of Sp(2n,2) ofdegrees 22n-1±2n-1. We construct two new infinite families of strongly incidence-transitive codes and demonstrate that they arethe only examples with codeword stabilisers contained in a geometric Aschbacher class. We construct two additional examplesusing the fully deleted permutation modules for the symmetric group Sm with m=10 and show no further examples arise forother values of m. If a codeword stabiliser is almost-simple then we show in most cases that the corresponding code cannot bestrongly incidence-transitive, though several possibilities remain open.",

keywords = "Jordan-Steiner actions, Codes in graphs, Johnson graphs, Quadratic forms, Symplectic geometry, Strongly incidence-transitive, Neighbour-transitive, Automorphism group",

author = "Mark Ioppolo",

year = "2019",

doi = "10.26182/5efd33ee8efef",

language = "English",

school = "The University of Western Australia",

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AU - Ioppolo, Mark

PY - 2019

Y1 - 2019

N2 - This thesis studies strongly incidence-transitive codes in Johnson graphs associated with the 2-transitive actions of Sp(2n,2) ofdegrees 22n-1±2n-1. We construct two new infinite families of strongly incidence-transitive codes and demonstrate that they arethe only examples with codeword stabilisers contained in a geometric Aschbacher class. We construct two additional examplesusing the fully deleted permutation modules for the symmetric group Sm with m=10 and show no further examples arise forother values of m. If a codeword stabiliser is almost-simple then we show in most cases that the corresponding code cannot bestrongly incidence-transitive, though several possibilities remain open.

AB - This thesis studies strongly incidence-transitive codes in Johnson graphs associated with the 2-transitive actions of Sp(2n,2) ofdegrees 22n-1±2n-1. We construct two new infinite families of strongly incidence-transitive codes and demonstrate that they arethe only examples with codeword stabilisers contained in a geometric Aschbacher class. We construct two additional examplesusing the fully deleted permutation modules for the symmetric group Sm with m=10 and show no further examples arise forother values of m. If a codeword stabiliser is almost-simple then we show in most cases that the corresponding code cannot bestrongly incidence-transitive, though several possibilities remain open.

KW - Jordan-Steiner actions

KW - Codes in graphs

KW - Johnson graphs

KW - Quadratic forms

KW - Symplectic geometry

KW - Strongly incidence-transitive

KW - Neighbour-transitive

KW - Automorphism group

U2 - 10.26182/5efd33ee8efef

DO - 10.26182/5efd33ee8efef

M3 - Doctoral Thesis

ER -

Codes in Johnson graphs associated with quadratic forms over F2

Abstract

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Codes in Johnson graphs associated with quadratic forms over F₂