MinT - Best Known (51, 51+55, s)-Nets in Base 27

Best Known (51, 51+55, s)-Nets in Base 27

(51, 51+55, 192)-Net over F₂₇ — Constructive and digital

Digital (51, 106, 192)-net over F₂₇, using

3 times m-reduction [i] based on digital (51, 109, 192)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 40, 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, 69, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(51, 51+55, 370)-Net in Base 27 — Constructive

(51, 106, 370)-net in base 27, using

t-expansion [i] based on (43, 106, 370)-net in base 27, using
- 2 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]

(51, 51+55, 520)-Net over F₂₇ — Digital

Digital (51, 106, 520)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+55, 154813)-Net in Base 27 — Upper bound on s

There is no (51, 106, 154814)-net in base 27, because

1 times m-reduction [i] would yield (51, 105, 154814)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 964432 443406 608320 033676 242844 303173 945133 266662 266975 388242 185230 557227 923558 124614 232385 140795 659042 523348 181736 939386 787383 348233 438343 821045 131361 > 27¹⁰⁵ [i]