MinT - Best Known (35, 60, s)-Nets in Base 27

Best Known (35, 60, s)-Nets in Base 27

(35, 60, 210)-Net over F₂₇ — Constructive and digital

Digital (35, 60, 210)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 12, 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 (4, 16, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
3. digital (7, 32, 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]

(35, 60, 370)-Net in Base 27 — Constructive

(35, 60, 370)-net in base 27, using

16 times m-reduction [i] based on (35, 76, 370)-net in base 27, using
- base change [i] based on digital (16, 57, 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]

(35, 60, 1440)-Net over F₂₇ — Digital

Digital (35, 60, 1440)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(35, 60, 2217815)-Net in Base 27 — Upper bound on s

There is no (35, 60, 2217816)-net in base 27, because

1 times m-reduction [i] would yield (35, 59, 2217816)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2 821391 591169 742920 917334 839967 688227 785589 517920 306546 918088 406047 484169 934374 269121 > 27⁵⁹ [i]