1. Tiling rings with “precious” differences
-
Marco Buratti
Abstract
We determine the maximum size of a difference packing of a ring R whose blocks are all multiples of {1, ϕ, ϕ2} with ϕ gold (i. e., a solution of x2 − x − 1 = 0 in R) or all multiples of {1, ψ, ψ2, ψ3} with ψ platinum (i. e., a solution of x3 + x2 − 1 = 0 in R). As a special consequence, we get a few more classes of values of v for which now we can claim that an optimal (v, 4, 1) optical orthogonal code exists. We think that this “golden” context is the right one for reformulating an old difference family construction of the present author (improving a much older one by R. C. Bose) in terms of golden elements of a field.
Abstract
We determine the maximum size of a difference packing of a ring R whose blocks are all multiples of {1, ϕ, ϕ2} with ϕ gold (i. e., a solution of x2 − x − 1 = 0 in R) or all multiples of {1, ψ, ψ2, ψ3} with ψ platinum (i. e., a solution of x3 + x2 − 1 = 0 in R). As a special consequence, we get a few more classes of values of v for which now we can claim that an optimal (v, 4, 1) optical orthogonal code exists. We think that this “golden” context is the right one for reformulating an old difference family construction of the present author (improving a much older one by R. C. Bose) in terms of golden elements of a field.
Kapitel in diesem Buch
- Frontmatter I
- Introduction V
- Contents VII
- 1. Tiling rings with “precious” differences 1
- 2. q-analogs of group divisible designs 21
- 3. A survey on bent functions and their duals 39
- 4. Construction of Bent functions using covering extended building sets 57
- 5. Results on permutation polynomials of shape xt + γTrqn/q(xd) 67
- 6. A new structure for difference matrices over abelian p-groups 79
- 7. Weil sums of binomials: properties, applications and open problems 109
- 8. A survey on iterations of mappings over finite fields 135
- 9. Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs and related geometric substructures 173
- 10. Pseudorandomness of large sets in finite fields 199
- 11. Point-plane incidences and some applications in positive characteristic 211
- 12. A survey of group invariant Butson matrices and their relation to generalized bent functions and various other objects 241
- 13. MRD codes: constructions and connections 255
- 14. Differential fault attack on hardware stream ciphers—a technical survey 287
- 15. Polynomials over finite fields: an index approach 319
Kapitel in diesem Buch
- Frontmatter I
- Introduction V
- Contents VII
- 1. Tiling rings with “precious” differences 1
- 2. q-analogs of group divisible designs 21
- 3. A survey on bent functions and their duals 39
- 4. Construction of Bent functions using covering extended building sets 57
- 5. Results on permutation polynomials of shape xt + γTrqn/q(xd) 67
- 6. A new structure for difference matrices over abelian p-groups 79
- 7. Weil sums of binomials: properties, applications and open problems 109
- 8. A survey on iterations of mappings over finite fields 135
- 9. Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs and related geometric substructures 173
- 10. Pseudorandomness of large sets in finite fields 199
- 11. Point-plane incidences and some applications in positive characteristic 211
- 12. A survey of group invariant Butson matrices and their relation to generalized bent functions and various other objects 241
- 13. MRD codes: constructions and connections 255
- 14. Differential fault attack on hardware stream ciphers—a technical survey 287
- 15. Polynomials over finite fields: an index approach 319
