MinT - Best Known (23, 23+45, s)-Nets in Base 27

Best Known (23, 23+45, s)-Nets in Base 27

(23, 23+45, 114)-Net over F₂₇ — Constructive and digital

Digital (23, 68, 114)-net over F₂₇, using

net from sequence [i] based on digital (23, 113)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 23 and N(F) ≥ 114, using
  - an explicitly constructive algebraic function field [i]

(23, 23+45, 160)-Net in Base 27 — Constructive

(23, 68, 160)-net in base 27, using

4 times m-reduction [i] based on (23, 72, 160)-net in base 27, using
- base change [i] based on digital (5, 54, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(23, 23+45, 163)-Net over F₂₇ — Digital

Digital (23, 68, 163)-net over F₂₇, using

t-expansion [i] based on digital (21, 68, 163)-net over F₂₇, using
- net from sequence [i] based on digital (21, 162)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 21 and N(F) ≥ 163, using
    - a curve from the manYPoints database [i]

(23, 23+45, 190)-Net in Base 27

(23, 68, 190)-net in base 27, using

base change [i] based on digital (6, 51, 190)-net over F₈₁, using
- net from sequence [i] based on digital (6, 189)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 6 and N(F) ≥ 190, using
    - a curve from the manYPoints database [i]

(23, 23+45, 7950)-Net in Base 27 — Upper bound on s

There is no (23, 68, 7951)-net in base 27, because

1 times m-reduction [i] would yield (23, 67, 7951)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 797160 610829 174880 769732 635203 747085 892709 794212 866107 140356 448892 391175 716787 108164 928779 592549 > 27⁶⁷ [i]