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Enumeration of Binary Linear Codes from the Orthogonal Extension Group O8+(2):2 Using Modular Representation Theory
Subject area: Science,Engineering and Technology · Area of research: Coding theory
Abstract
This paper presents a comprehensive enumeration of binary linear codes constructed from maximal subgroups of the orthogonal extension group O??(2):2. Using modular representation theory and computational methods in MAGMA, we systematically analyze three distinct permutation representations of degrees 120, 135, and 960. The enumeration reveals 162 total submodules across the first two representations, yielding 8 featured binary linear codes with parameters ranging from [120,8,56]? to [135,35,27]?. Notable findings include doubly even codes with exceptional minimum distances, projective codes with superior error-correction capabilities, and irreducible codes demonstrating optimal structural properties. The 120-dimensional representation produces codes generating primitive combinatorial designs, while the 135-dimensional representation yields codes with enhanced error-detecting capabilities. These results establish O??(2):2 as a rich source of high-quality linear codes for cryptographic and communication applications.
Keywords
Orthogonal groups, Extension groups, Linear codes, Modular representation
How to cite this paper
@article{1710425,
author = {Janet Lilian Maina, Vincent Marani},
title = {Enumeration of Binary Linear Codes from the Orthogonal Extension Group O8+(2):2 Using Modular Representation Theory},
journal = {Iconic Research And Engineering Journals},
year = {2025},
volume = {9},
number = {3},
pages = {1025-1029},
issn = {2456-8880},
url = {https://www.irejournals.com/formatedpaper/1710425.pdf},
abstract = {This paper presents a comprehensive enumeration of binary linear codes constructed from maximal subgroups of the orthogonal extension group O??(2):2. Using modular representation theory and computational methods in MAGMA, we systematically analyze three distinct permutation representations of degrees 120, 135, and 960. The enumeration reveals 162 total submodules across the first two representations, yielding 8 featured binary linear codes with parameters ranging from [120,8,56]? to [135,35,27]?. Notable findings include doubly even codes with exceptional minimum distances, projective codes with superior error-correction capabilities, and irreducible codes demonstrating optimal structural properties. The 120-dimensional representation produces codes generating primitive combinatorial designs, while the 135-dimensional representation yields codes with enhanced error-detecting capabilities. These results establish O??(2):2 as a rich source of high-quality linear codes for cryptographic and communication applications.},
keywords = {Orthogonal groups, Extension groups, Linear codes, Modular representation},
month = {September},
}