MinT - Best Known (61, 61+34, s)-Nets in Base 27

Best Known (61, 61+34, s)-Nets in Base 27

(61, 61+34, 312)-Net over F₂₇ — Constructive and digital

Digital (61, 95, 312)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 9, 56)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
        the rational function field F₂₇(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 6, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
2. digital (4, 12, 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]
3. digital (4, 15, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
4. digital (4, 21, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
5. digital (4, 38, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(61, 61+34, 730)-Net in Base 27 — Constructive

(61, 95, 730)-net in base 27, using

5 times m-reduction [i] based on (61, 100, 730)-net in base 27, using
- base change [i] based on digital (36, 75, 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]

(61, 61+34, 6699)-Net over F₂₇ — Digital

Digital (61, 95, 6699)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(61, 61+34, large)-Net in Base 27 — Upper bound on s

There is no (61, 95, large)-net in base 27, because

32 times m-reduction [i] would yield (61, 63, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]