MinT - Best Known (47, 47+47, s)-Nets in Base 27

Best Known (47, 47+47, s)-Nets in Base 27

(47, 47+47, 192)-Net over F₂₇ — Constructive and digital

Digital (47, 94, 192)-net over F₂₇, using

3 times m-reduction [i] based on digital (47, 97, 192)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 36, 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, 61, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(47, 47+47, 370)-Net in Base 27 — Constructive

(47, 94, 370)-net in base 27, using

t-expansion [i] based on (43, 94, 370)-net in base 27, using
- 14 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]

(47, 47+47, 596)-Net over F₂₇ — Digital

Digital (47, 94, 596)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(47, 47+47, 222406)-Net in Base 27 — Upper bound on s

There is no (47, 94, 222407)-net in base 27, because

1 times m-reduction [i] would yield (47, 93, 222407)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 13 087018 441514 148147 224709 759112 515248 342921 832035 979761 097242 714110 989569 297382 768403 000810 928981 167128 316254 756232 635254 811332 715915 > 27⁹³ [i]