MinT - Best Known (51, 51+50, s)-Nets in Base 27

Best Known (51, 51+50, s)-Nets in Base 27

(51, 51+50, 192)-Net over F₂₇ — Constructive and digital

Digital (51, 101, 192)-net over F₂₇, using

8 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+50, 370)-Net in Base 27 — Constructive

(51, 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]

(51, 51+50, 670)-Net over F₂₇ — Digital

Digital (51, 101, 670)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+50, 237322)-Net in Base 27 — Upper bound on s

There is no (51, 101, 237323)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3 696380 910332 948581 073129 702766 621093 098223 815596 339230 660617 127820 307400 501334 803240 182302 844613 975371 984054 159467 179456 985933 402493 885746 151919 > 27¹⁰¹ [i]