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- Title
Invariant codes, difference schemes, and distributive quasigroups.
- Authors
Castoldi, André Guerino; Martinhão, Anderson Novaes; MonteCarmelo, Emerson L.; dos Santos, Otávio J. N. T. N.
- Abstract
Regard T = { (g , ... , g) ∈ F q n : g ∈ F q } as an additive subgroup of the vector space F q n over a finite field F q . A code C ⊂ F q n is T-invariant if T acts on C induced by the addition of F q n . Using linear algebra and finite fields results, we investigate which important families of codes are T-invariant, including perfect codes, first-order Reed–Muller codes, and some Reed–Solomon codes. In particular, we characterize which perfect linear codes are T-invariant. A version of the Gilbert–Varshamov theorem for T-invariant codes is presented. We characterize which Hadamard codes are T-invariant for nonlinear codes. As an application, constructions of difference schemes are obtained using our results. Finally, difference schemes of strength 2 over a finite field are characterized using quasigroups endowed with a distributive law.
- Subjects
HADAMARD codes; REED-Solomon codes; QUASIGROUPS; LINEAR codes; REED-Muller codes; LINEAR algebra; FINITE fields; VECTOR spaces; MAGIC squares
- Publication
Computational & Applied Mathematics, 2022, Vol 41, Issue 8, p1
- ISSN
0101-8205
- Publication type
Article
- DOI
10.1007/s40314-022-02069-w