MinT - Best Known (105−41, 105, s)-Nets in Base 27

Best Known (105−41, 105, s)-Nets in Base 27

(105−41, 105, 286)-Net over F₂₇ — Constructive and digital

Digital (64, 105, 286)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 14, 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 (4, 17, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
3. digital (6, 26, 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]
4. digital (7, 48, 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]

(105−41, 105, 730)-Net in Base 27 — Constructive

(64, 105, 730)-net in base 27, using

t-expansion [i] based on (63, 105, 730)-net in base 27, using
- 3 times m-reduction [i] based on (63, 108, 730)-net in base 27, using
  - base change [i] based on digital (36, 81, 730)-net over F₈₁, using
    - net from sequence [i] based on digital (36, 729)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 36 and N(F) ≥ 730, using
        the Hermitian function field over F₈₁ [i]

(105−41, 105, 3489)-Net over F₂₇ — Digital

Digital (64, 105, 3489)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(105−41, 105, large)-Net in Base 27 — Upper bound on s

There is no (64, 105, large)-net in base 27, because

39 times m-reduction [i] would yield (64, 66, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]