MinT - Best Known (50, 83, s)-Nets in Base 27

Best Known (50, 83, s)-Nets in Base 27

(50, 83, 252)-Net over F₂₇ — Constructive and digital

Digital (50, 83, 252)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 18, 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]
2. digital (7, 23, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (9, 42, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(50, 83, 408)-Net in Base 27 — Constructive

(50, 83, 408)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 38)-net over F₂₇, using
  - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. (33, 66, 370)-net in base 27, using
  - 2 times m-reduction [i] based on (33, 68, 370)-net in base 27, using
    - base change [i] based on digital (16, 51, 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]

(50, 83, 2554)-Net over F₂₇ — Digital

Digital (50, 83, 2554)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 83, 5666358)-Net in Base 27 — Upper bound on s

There is no (50, 83, 5666359)-net in base 27, because

1 times m-reduction [i] would yield (50, 82, 5666359)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2354 123853 179371 261173 754713 651147 866392 396823 734558 121589 391615 856498 769343 957036 920286 227278 419650 316017 090353 723745 > 27⁸² [i]