MinT - Best Known (23, 23+59, s)-Nets in Base 27

Best Known (23, 23+59, s)-Nets in Base 27

(23, 23+59, 114)-Net over F₂₇ — Constructive and digital

Digital (23, 82, 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+59, 116)-Net in Base 27 — Constructive

(23, 82, 116)-net in base 27, using

2 times m-reduction [i] based on (23, 84, 116)-net in base 27, using
- base change [i] based on digital (2, 63, 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]

(23, 23+59, 163)-Net over F₂₇ — Digital

Digital (23, 82, 163)-net over F₂₇, using

t-expansion [i] based on digital (21, 82, 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+59, 4452)-Net in Base 27 — Upper bound on s

There is no (23, 82, 4453)-net in base 27, because

1 times m-reduction [i] would yield (23, 81, 4453)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 87 350017 689108 161311 632377 639888 430685 747271 089514 529512 831937 195298 329693 356072 449185 461028 352773 190676 131648 244115 > 27⁸¹ [i]