MinT - Best Known (105−61, 105, s)-Nets in Base 27

Best Known (105−61, 105, s)-Nets in Base 27

(105−61, 105, 164)-Net over F₂₇ — Constructive and digital

Digital (44, 105, 164)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 37, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]
2. digital (7, 68, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(105−61, 105, 280)-Net over F₂₇ — Digital

Digital (44, 105, 280)-net over F₂₇, using

t-expansion [i] based on digital (42, 105, 280)-net over F₂₇, using
- net from sequence [i] based on digital (42, 279)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 42 and N(F) ≥ 280, using
    - Drinfeld modules of rank 1 [i]

(105−61, 105, 370)-Net in Base 27 — Constructive

(44, 105, 370)-net in base 27, using

t-expansion [i] based on (43, 105, 370)-net in base 27, using
- 3 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(105−61, 105, 42434)-Net in Base 27 — Upper bound on s

There is no (44, 105, 42435)-net in base 27, because

1 times m-reduction [i] would yield (44, 104, 42435)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 72789 960439 275324 240728 977508 352078 918920 747302 996216 718677 571248 848967 414233 305884 003209 465817 154899 097773 687036 335783 000047 389605 409228 143658 244317 > 27¹⁰⁴ [i]