MinT - Best Known (58−37, 58, s)-Nets in Base 27

Best Known (58−37, 58, s)-Nets in Base 27

(58−37, 58, 108)-Net over F₂₇ — Constructive and digital

Digital (21, 58, 108)-net over F₂₇, using

t-expansion [i] based on digital (18, 58, 108)-net over F₂₇, using
- net from sequence [i] based on digital (18, 107)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 108, using
    - F₃ from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F₂₇ [i]

(58−37, 58, 160)-Net in Base 27 — Constructive

(21, 58, 160)-net in base 27, using

6 times m-reduction [i] based on (21, 64, 160)-net in base 27, using
- base change [i] based on digital (5, 48, 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]

(58−37, 58, 163)-Net over F₂₇ — Digital

Digital (21, 58, 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]

(58−37, 58, 190)-Net in Base 27

(21, 58, 190)-net in base 27, using

2 times m-reduction [i] based on (21, 60, 190)-net in base 27, using
- base change [i] based on digital (6, 45, 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]

(58−37, 58, 9894)-Net in Base 27 — Upper bound on s

There is no (21, 58, 9895)-net in base 27, because

1 times m-reduction [i] would yield (21, 57, 9895)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3871 870671 554960 940121 472025 385195 581275 132509 620997 156999 642581 578974 472525 120253 > 27⁵⁷ [i]