MinT - Best Known (68, 68+42, s)-Nets in Base 27

Best Known (68, 68+42, s)-Nets in Base 27

(68, 68+42, 298)-Net over F₂₇ — Constructive and digital

Digital (68, 110, 298)-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 (6, 20, 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 (6, 27, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
4. digital (7, 49, 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]

(68, 68+42, 730)-Net in Base 27 — Constructive

(68, 110, 730)-net in base 27, using

27² times duplication [i] based on (66, 108, 730)-net in base 27, using
- t-expansion [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]

(68, 68+42, 4318)-Net over F₂₇ — Digital

Digital (68, 110, 4318)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+42, large)-Net in Base 27 — Upper bound on s

There is no (68, 110, large)-net in base 27, because

40 times m-reduction [i] would yield (68, 70, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]