MinT - Best Known (36, 36+37, s)-Nets in Base 27

Best Known (36, 36+37, s)-Nets in Base 27

(36, 36+37, 178)-Net over F₂₇ — Constructive and digital

Digital (36, 73, 178)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (8, 26, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]
2. digital (10, 47, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(36, 36+37, 370)-Net in Base 27 — Constructive

(36, 73, 370)-net in base 27, using

7 times m-reduction [i] based on (36, 80, 370)-net in base 27, using
- base change [i] based on digital (16, 60, 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]

(36, 36+37, 455)-Net over F₂₇ — Digital

Digital (36, 73, 455)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(36, 36+37, 154378)-Net in Base 27 — Upper bound on s

There is no (36, 73, 154379)-net in base 27, because

1 times m-reduction [i] would yield (36, 72, 154379)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 11 434909 091434 376790 687290 471463 256221 652300 971107 331755 184925 852027 624983 430453 579376 785292 799146 567269 > 27⁷² [i]