MinT - Best Known (78−41, 78, s)-Nets in Base 27

Best Known (78−41, 78, s)-Nets in Base 27

(78−41, 78, 176)-Net over F₂₇ — Constructive and digital

Digital (37, 78, 176)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 27, 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 (10, 51, 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]

(78−41, 78, 370)-Net in Base 27 — Constructive

(37, 78, 370)-net in base 27, using

6 times m-reduction [i] based on (37, 84, 370)-net in base 27, using
- base change [i] based on digital (16, 63, 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]

(78−41, 78, 380)-Net over F₂₇ — Digital

Digital (37, 78, 380)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−41, 78, 103523)-Net in Base 27 — Upper bound on s

There is no (37, 78, 103524)-net in base 27, because

1 times m-reduction [i] would yield (37, 77, 103524)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 164 068335 746498 123210 333719 025673 380241 741470 399476 928829 213494 604965 943584 906707 727698 671323 379352 396558 875201 > 27⁷⁷ [i]