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Superregular Matrices over Finite Fields

Author

Listed:
  • Paulo Almeida

    (CIDMA—Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal)

  • Miguel Beltrá

    (Department of Mathematics, University of Alicante, 03690 San Vicente del Raspeig, Spain)

  • Diego Napp

    (Department of Mathematics, University of Alicante, 03690 San Vicente del Raspeig, Spain)

Abstract

A trivially zero minor of a matrix is a minor having all its terms in the Leibniz formula equal to zero. A matrix is superregular if all of its minors that are not trivially zero are nonzero. In the area of Coding Theory, superregular matrices over finite fields are connected with codes with optimum error correcting capabilities. There are two types of superregular matrices that yield two different types of codes. One has in all of its entries a nonzero element, and these are called full superregular matrices. The second interesting class of superregular matrices is formed by lower triangular Toeplitz matrices. In contrast to full superregular matrices, all general constructions of these matrices require very large field sizes. In this work, we investigate the construction of lower triangular Toeplitz superregular matrices over small finite prime fields. Instead of computing all possible minors, we study the structure of finite fields in order to reduce the possible nonzero minors. This allows us to restrict the huge number of possibilities that one needs to check and come up with novel constructions of superregular matrices over relatively small fields. Finally, we present concrete examples of lower triangular Toeplitz superregular matrices of sizes up to 10.

Suggested Citation

  • Paulo Almeida & Miguel Beltrá & Diego Napp, 2025. "Superregular Matrices over Finite Fields," Mathematics, MDPI, vol. 13(7), pages 1-23, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:7:p:1091-:d:1621150
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    References listed on IDEAS

    as
    1. Marija Delić & Jelena Ivetić, 2025. "On the Maximum Probability of Full Rank of Random Matrices over Finite Fields," Mathematics, MDPI, vol. 13(3), pages 1-8, February.
    2. Adel N. Alahmadi & Husain S. Alhazmi & Hatoon Shoaib & David G. Glynn & Saeed Ur Rehman & Patrick Solé, 2023. "Connections between Linear Complementary Dual Codes, Permanents and Geometry," Mathematics, MDPI, vol. 11(12), pages 1-11, June.
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