MinT - Best Known (36, 87, s)-Nets in Base 27

Best Known (36, 87, s)-Nets in Base 27

(36, 87, 146)-Net over F₂₇ — Constructive and digital

Digital (36, 87, 146)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 29, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (7, 58, 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]

(36, 87, 224)-Net in Base 27 — Constructive

(36, 87, 224)-net in base 27, using

5 times m-reduction [i] based on (36, 92, 224)-net in base 27, using
- base change [i] based on digital (13, 69, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(36, 87, 244)-Net over F₂₇ — Digital

Digital (36, 87, 244)-net over F₂₇, using

net from sequence [i] based on digital (36, 243)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 36 and N(F) ≥ 244, using
  - Drinfeld modules of rank 1 [i]

(36, 87, 298)-Net in Base 27

(36, 87, 298)-net in base 27, using

9 times m-reduction [i] based on (36, 96, 298)-net in base 27, using
- base change [i] based on digital (12, 72, 298)-net over F₈₁, using
  - net from sequence [i] based on digital (12, 297)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 12 and N(F) ≥ 298, using
      - a curve from the manYPoints database [i]

(36, 87, 32837)-Net in Base 27 — Upper bound on s

There is no (36, 87, 32838)-net in base 27, because

1 times m-reduction [i] would yield (36, 86, 32838)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1251 381650 148370 049013 181408 771457 319300 893569 609407 713411 935699 711371 703379 120827 694485 218354 321458 989587 385193 863688 942445 > 27⁸⁶ [i]