Connections between Linear Complementary Dual Codes, Permanents and Geometry

Adel N. Alahmadi (), Husain S. Alhazmi, Hatoon Shoaib, David G. Glynn, Saeed Ur Rehman and Patrick Solé
Additional contact information
Adel N. Alahmadi: Research Group of Algebraic Structures and Applications (ASA), Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Husain S. Alhazmi: Research Group of Algebraic Structures and Applications (ASA), Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Hatoon Shoaib: Research Group of Algebraic Structures and Applications (ASA), Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
David G. Glynn: College of Science and Engineering, Flinders University, G.P.O. Box 2100, Tonsley, SA 5001, Australia
Saeed Ur Rehman: College of Science and Engineering, Flinders University, G.P.O. Box 2100, Tonsley, SA 5001, Australia
Patrick Solé: 12M Lab, (Centrale Marseille, CNRS, Aix-Marseille University), 13288 Marseilles, France

Mathematics, 2023, vol. 11, issue 12, 1-11

Abstract: Linear codes with complementary duals, or LCD codes, have recently been applied to side-channel and fault injection attack-resistant cryptographic countermeasures. We explain that over characteristic two fields, they exist whenever the permanent of any generator matrix is non-zero. Alternatively, in the binary case, the matroid represented by the columns of the matrix has an odd number of bases. We explain how Grassmannian varieties as well as linear and quadratic complexes are connected with LCD codes. Accessing the classification of polarities, we relate the binary LCD codes of dimension k to the two kinds of symmetric non-singular binary matrices, to certain truncated Reed–Muller codes, and to the geometric codes of planes in finite projective space via the self-orthogonal codes of dimension k .

Keywords: error-correcting code; invariant; rectangular matrix; permanent; complementary code; geometric code; Reed–Muller; Grassmannian; polarity; self-orthogonal; matroid (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
https://www.mdpi.com/2227-7390/11/12/2774/pdf (application/pdf)
https://www.mdpi.com/2227-7390/11/12/2774/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:11:y:2023:i:12:p:2774-:d:1174806

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().