Connections between Linear Complementary Dual Codes, Permanents and Geometry
Adel N. Alahmadi,
Husain S. Alhazmi,
Hatoon Shoaib,
David G. Glynn,
Saeed Ur Rehman,
Patrick Solé
Affiliations
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
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.
