MinT - Best Known (76−31, 76, s)-Nets in Base 27

Best Known (76−31, 76, s)-Nets in Base 27

(76−31, 76, 240)-Net over F₂₇ — Constructive and digital

Digital (45, 76, 240)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 16, 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]
2. digital (7, 22, 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]
3. digital (7, 38, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(76−31, 76, 370)-Net in Base 27 — Constructive

(45, 76, 370)-net in base 27, using

t-expansion [i] based on (43, 76, 370)-net in base 27, using
- 32 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]

(76−31, 76, 1974)-Net over F₂₇ — Digital

Digital (45, 76, 1974)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(76−31, 76, 3544958)-Net in Base 27 — Upper bound on s

There is no (45, 76, 3544959)-net in base 27, because

1 times m-reduction [i] would yield (45, 75, 3544959)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 225051 950618 881925 731614 320389 092642 465203 389551 573786 732626 063386 551083 175594 338201 779861 128816 842981 640315 > 27⁷⁵ [i]