MinT - Best Known (101−51, 101, s)-Nets in Base 27

Best Known (101−51, 101, s)-Nets in Base 27

(101−51, 101, 192)-Net over F₂₇ — Constructive and digital

Digital (50, 101, 192)-net over F₂₇, using

5 times m-reduction [i] based on digital (50, 106, 192)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 39, 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, 67, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(101−51, 101, 370)-Net in Base 27 — Constructive

(50, 101, 370)-net in base 27, using

t-expansion [i] based on (43, 101, 370)-net in base 27, using
- 7 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]

(101−51, 101, 591)-Net over F₂₇ — Digital

Digital (50, 101, 591)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(101−51, 101, 208008)-Net in Base 27 — Upper bound on s

There is no (50, 101, 208009)-net in base 27, because

1 times m-reduction [i] would yield (50, 100, 208009)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 136904 489968 195685 467070 856597 753020 519589 083148 276689 384377 935464 996796 275374 781168 634021 541538 790895 340566 176101 594816 348907 310574 602061 375307 > 27¹⁰⁰ [i]