MinT - Best Known (13, 13+29, s)-Nets in Base 27

Best Known (13, 13+29, s)-Nets in Base 27

(13, 13+29, 96)-Net over F₂₇ — Constructive and digital

Digital (13, 42, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 42, 96)-net over F₂₇, using
- net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
    - an explicitly constructive algebraic function field [i]

(13, 13+29, 116)-Net in Base 27 — Constructive

(13, 42, 116)-net in base 27, using

2 times m-reduction [i] based on (13, 44, 116)-net in base 27, using
- base change [i] based on digital (2, 33, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(13, 13+29, 136)-Net over F₂₇ — Digital

Digital (13, 42, 136)-net over F₂₇, using

net from sequence [i] based on digital (13, 135)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 13 and N(F) ≥ 136, using
  - a curve from the manYPoints database [i]

(13, 13+29, 3609)-Net in Base 27 — Upper bound on s

There is no (13, 42, 3610)-net in base 27, because

1 times m-reduction [i] would yield (13, 41, 3610)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 48546 377871 734994 102687 095161 883598 205257 082455 009636 988893 > 27⁴¹ [i]