MinT - Best Known (69, 109, s)-Nets in Base 27

Best Known (69, 109, s)-Nets in Base 27

(69, 109, 316)-Net over F₂₇ — Constructive and digital

Digital (69, 109, 316)-net over F₂₇, using

1 times m-reduction [i] based on digital (69, 110, 316)-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 (6, 19, 76)-net over F₂₇, using
    - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
  3. digital (7, 27, 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]
  4. digital (7, 48, 82)-net over F₂₇, using
    - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(69, 109, 730)-Net in Base 27 — Constructive

(69, 109, 730)-net in base 27, using

27¹ times duplication [i] based on (68, 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]

(69, 109, 5948)-Net over F₂₇ — Digital

Digital (69, 109, 5948)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 109, large)-Net in Base 27 — Upper bound on s

There is no (69, 109, large)-net in base 27, because

38 times m-reduction [i] would yield (69, 71, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]