MinT - Best Known (85−42, 85, s)-Nets in Base 27

Best Known (85−42, 85, s)-Nets in Base 27

(85−42, 85, 192)-Net over F₂₇ — Constructive and digital

Digital (43, 85, 192)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (11, 32, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
      - an explicitly constructive algebraic function field [i]
2. digital (11, 53, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(85−42, 85, 370)-Net in Base 27 — Constructive

(43, 85, 370)-net in base 27, using

23 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]

(85−42, 85, 597)-Net over F₂₇ — Digital

Digital (43, 85, 597)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(85−42, 85, 207544)-Net in Base 27 — Upper bound on s

There is no (43, 85, 207545)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 46 337124 617908 068180 575440 106494 784162 187884 617408 486511 685889 868370 427036 662486 196133 579539 511059 592437 054340 758145 581787 > 27⁸⁵ [i]