MinT - Best Known (84−37, 84, s)-Nets in Base 27

Best Known (84−37, 84, s)-Nets in Base 27

(84−37, 84, 222)-Net over F₂₇ — Constructive and digital

Digital (47, 84, 222)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 16, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (6, 24, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
3. digital (7, 44, 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]

(84−37, 84, 370)-Net in Base 27 — Constructive

(47, 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]

(84−37, 84, 1219)-Net over F₂₇ — Digital

Digital (47, 84, 1219)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(84−37, 84, 1156992)-Net in Base 27 — Upper bound on s

There is no (47, 84, 1156993)-net in base 27, because

1 times m-reduction [i] would yield (47, 83, 1156993)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 63561 555460 990877 450676 898582 724576 263852 114082 785392 157204 364448 452095 593430 931987 488324 995745 474919 220752 136932 081049 > 27⁸³ [i]