MinT - Best Known (110−51, 110, s)-Nets in Base 27

Best Known (110−51, 110, s)-Nets in Base 27

(110−51, 110, 222)-Net over F₂₇ — Constructive and digital

Digital (59, 110, 222)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 21, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (6, 31, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
3. digital (7, 58, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(110−51, 110, 370)-Net in Base 27 — Constructive

(59, 110, 370)-net in base 27, using

27² times duplication [i] based on (57, 108, 370)-net in base 27, using
- t-expansion [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]

(110−51, 110, 1081)-Net over F₂₇ — Digital

Digital (59, 110, 1081)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(110−51, 110, 681381)-Net in Base 27 — Upper bound on s

There is no (59, 110, 681382)-net in base 27, because

1 times m-reduction [i] would yield (59, 109, 681382)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 043904 975845 878471 662471 322005 344239 049210 370245 331493 450233 545329 331323 875987 233794 866311 815625 601559 943422 645955 821331 752487 901751 629981 150967 601813 014317 > 27¹⁰⁹ [i]