MinT - Best Known (79−41, 79, s)-Nets in Base 27

Best Known (79−41, 79, s)-Nets in Base 27

(79−41, 79, 178)-Net over F₂₇ — Constructive and digital

Digital (38, 79, 178)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (8, 28, 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, 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]

(79−41, 79, 370)-Net in Base 27 — Constructive

(38, 79, 370)-net in base 27, using

9 times m-reduction [i] based on (38, 88, 370)-net in base 27, using
- base change [i] based on digital (16, 66, 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]

(79−41, 79, 415)-Net over F₂₇ — Digital

Digital (38, 79, 415)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(79−41, 79, 122071)-Net in Base 27 — Upper bound on s

There is no (38, 79, 122072)-net in base 27, because

1 times m-reduction [i] would yield (38, 78, 122072)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 4429 902757 980495 263201 744671 530897 647535 644766 268443 458107 063283 205161 490761 692689 510156 160057 922243 631675 980481 > 27⁷⁸ [i]