Linear Codes from Modular Representations of the General Linear Groups GL(2,7) and GL(2,8)

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@article{1720307,
      author = {Vincent Nyongesa Marani},
      title = {Linear Codes from Modular Representations of the General Linear Groups GL(2,7) and GL(2,8)},
      journal = {Iconic Research And Engineering Journals},
      year = {2026},
      volume = {10},
      number = {2},
      pages = {3714-3721},
      issn = {2456-8880},
      url = {https://www.irejournals.com/formatedpaper/1720307.pdf},
      abstract = {This paper constructs and classifies the linear codes arising as submodules of the permutation modules of the general linear groups GL(2,7) and GL(2,8) acting on the cosets of their maximal subgroups, over every prime field GF(p) with p dividing the group order. Using the MeatAxe to determine complete submodule lattices and a tiered, cross-validated pipeline for minimum-distance computation - the Brouwer-Zimmermann algorithm for low dimensions and dual-code enumeration with the MacWilliams transform for high dimensions - a total of 163 distinct codes is obtained: 106 from the three primitive actions of GL(2,7) (degrees 8, 21 and 28) and 71 from the four primitive actions of GL(2,8) (degrees 7, 9, 28 and 36). The constructions recover classical objects, including the binary Hamming code [7,4,3] and its simplex dual [7,3,4], together with a complete chain of MDS codes [7,k,8-k] over GF(7) for 1 ≤ k ≤ 6, and produce codes of small Singleton defect such as [21,14,6], [21,15,5] and [28,23,4] over GF(7). Structural phenomena of the underlying modules are documented, including the sharp contrast between defining- and cross-characteristic submodule lattices. All computations were performed in GAP with the GUAVA package, and complete scripts accompany the paper for independent verification.},
      keywords = {General linear group, linear code, MeatAxe, minimum distance, modular representation, permutation module.},
      month = {August},
      doi = {https://doi.org/10.64388/IREV10I2-1720307}
  }