MinT - Best Known (84−35, 84, s)-Nets in Base 27

Best Known (84−35, 84, s)-Nets in Base 27

(84−35, 84, 246)-Net over F₂₇ — Constructive and digital

Digital (49, 84, 246)-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, 24, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (7, 42, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(84−35, 84, 370)-Net in Base 27 — Constructive

(49, 84, 370)-net in base 27, using

t-expansion [i] based on (43, 84, 370)-net in base 27, using
- 24 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 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]

(84−35, 84, 1807)-Net over F₂₇ — Digital

Digital (49, 84, 1807)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(84−35, 84, 2687778)-Net in Base 27 — Upper bound on s

There is no (49, 84, 2687779)-net in base 27, because

1 times m-reduction [i] would yield (49, 83, 2687779)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 63561 279397 402651 710701 430951 471650 733062 774400 753756 849107 053218 217517 089405 187694 454221 156501 441827 666005 140633 248367 > 27⁸³ [i]